Left Termination proof of /tmp/tmp5LfWqo/mergesort

Left Termination of the query pattern mergesort_in_2(g, a) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇐)

↳2 Prolog

    ↳3 PrologToPiTRSProof (⇐)

        ↳4 PiTRS

        ↳5 DependencyPairsProof (⇔)

        ↳6 PiDP

        ↳7 DependencyGraphProof (⇔)

        ↳8 AND

            ↳9 PiDP

                ↳10 UsableRulesProof (⇔)

                ↳11 PiDP

                ↳12 PiDPToQDPProof (⇐)

                ↳13 QDP

                ↳14 NonTerminationProof (⇔)

                ↳15 NO

            ↳16 PiDP

                ↳17 UsableRulesProof (⇔)

                ↳18 PiDP

                ↳19 PiDPToQDPProof (⇐)

                ↳20 QDP

                ↳21 NonTerminationProof (⇔)

                ↳22 NO

            ↳23 PiDP

                ↳24 UsableRulesProof (⇔)

                ↳25 PiDP

                ↳26 PiDPToQDPProof (⇐)

                ↳27 QDP

                ↳28 Narrowing (⇐)

                ↳29 QDP

                ↳30 Narrowing (⇐)

                ↳31 QDP

                ↳32 NonTerminationProof (⇔)

                ↳33 NO

            ↳34 PiDP

                ↳35 UsableRulesProof (⇔)

                ↳36 PiDP

                ↳37 PiDPToQDPProof (⇐)

                ↳38 QDP

                ↳39 QDPSizeChangeProof (⇔)

                ↳40 YES

            ↳41 PiDP

                ↳42 UsableRulesProof (⇔)

                ↳43 PiDP

                ↳44 PiDPToQDPProof (⇐)

                ↳45 QDP

                ↳46 MRRProof (⇔)

                ↳47 QDP

                ↳48 PisEmptyProof (⇔)

                ↳49 YES

    ↳50 PrologToPiTRSProof (⇐)

        ↳51 PiTRS

        ↳52 DependencyPairsProof (⇔)

        ↳53 PiDP

        ↳54 DependencyGraphProof (⇔)

        ↳55 AND

            ↳56 PiDP

                ↳57 UsableRulesProof (⇔)

                ↳58 PiDP

                ↳59 PiDPToQDPProof (⇐)

                ↳60 QDP

                ↳61 NonTerminationProof (⇔)

                ↳62 NO

            ↳63 PiDP

                ↳64 UsableRulesProof (⇔)

                ↳65 PiDP

                ↳66 PiDPToQDPProof (⇐)

                ↳67 QDP

                ↳68 NonTerminationProof (⇔)

                ↳69 NO

            ↳70 PiDP

                ↳71 UsableRulesProof (⇔)

                ↳72 PiDP

                ↳73 PiDPToQDPProof (⇐)

                ↳74 QDP

                ↳75 Narrowing (⇐)

                ↳76 QDP

                ↳77 Narrowing (⇐)

                ↳78 QDP

                ↳79 NonTerminationProof (⇔)

                ↳80 NO

            ↳81 PiDP

                ↳82 UsableRulesProof (⇔)

                ↳83 PiDP

                ↳84 PiDPToQDPProof (⇐)

                ↳85 QDP

                ↳86 QDPSizeChangeProof (⇔)

                ↳87 YES

            ↳88 PiDP

                ↳89 UsableRulesProof (⇔)

                ↳90 PiDP

                ↳91 PiDPToQDPProof (⇐)

                ↳92 QDP

                ↳93 QDPOrderProof (⇔)

                ↳94 QDP

                ↳95 DependencyGraphProof (⇔)

                ↳96 QDP

                ↳97 UsableRulesProof (⇔)

                ↳98 QDP

                ↳99 QReductionProof (⇔)

                ↳100 QDP

                ↳101 QDPOrderProof (⇔)

                ↳102 QDP

                ↳103 DependencyGraphProof (⇔)

                ↳104 TRUE

(0) Obligation:

Clauses:

mergesort([], []).
mergesort(.(X, []), .(X, [])).
mergesort(.(X, .(Y, L1)), L2) :- ','(split2(.(X, .(Y, L1)), L3, L4), ','(mergesort(L3, L5), ','(mergesort(L4, L6), merge(L5, L6, L2)))).
split(L1, L2, L3) :- split0(L1, L2, L3).
split(L1, L2, L3) :- split1(L1, L2, L3).
split(L1, L2, L3) :- split2(L1, L2, L3).
split0([], [], []).
split1(.(X, []), .(X, []), []).
split2(.(X, .(Y, L1)), .(X, L2), .(Y, L3)) :- split(L1, L2, L3).
merge([], L1, L1).
merge(L1, [], L1).
merge(.(X, L1), .(Y, L2), .(X, L3)) :- ','(le(X, Y), merge(L1, .(Y, L2), L3)).
merge(.(X, L1), .(Y, L2), .(Y, L3)) :- ','(gt(X, Y), merge(.(X, L1), L2, L3)).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(X), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(Y)).
le(0, 0).

Queries:

mergesort(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: mergesort(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: mergesort(T1, T2), SCOPE: 1, CLAUSE: 0 | mergesort(T1, T2), SCOPE: 1, CLAUSE: 1 | mergesort(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | mergesort([], T2), SCOPE: 1, CLAUSE: 1 | mergesort([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: mergesort(T1, T2), SCOPE: 1, CLAUSE: 1 | mergesort(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nmergesort(T1, T2) !=? mergesort([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: mergesort([], T2), SCOPE: 1, CLAUSE: 1 | mergesort([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: mergesort([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: true | mergesort(.(T4, []), T2), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: mergesort(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nmergesort(T1, T2) !=? mergesort([], [])\nmergesort(T1, T2) !=? mergesort(.(X7, []), .(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: mergesort(.(T4, []), T2), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(split2(.(T10, .(T11, T12)), X21, X22), ','(mergesort(X21, X23), ','(mergesort(X22, X24), merge(X23, X24, T14))))\n\nT10 is ground\nT11 is ground\nT12 is ground\nX21 is free; \nX22 is free; \nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(split2(.(T10, .(T11, T12)), X21, X22), ','(mergesort(X21, X23), ','(mergesort(X22, X24), merge(X23, X24, T14)))), SCOPE: 2, CLAUSE: 8\n\nT10 is ground\nT11 is ground\nT12 is ground\nX21 is free; \nX22 is free; \nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(split(T24, X53, X54), ','(mergesort(.(T22, X53), X23), ','(mergesort(.(T23, X54), X24), merge(X23, X24, T14))))\n\nT22 is ground\nT23 is ground\nT24 is ground\nX23 is free; \nX24 is free; \nX53 is free; \nX54 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: split(T24, X53, X54)\n\nT24 is ground\nX53 is free; \nX54 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(mergesort(.(T22, T26), X23), ','(mergesort(.(T23, T27), X24), merge(X23, X24, T14)))\n\nT22 is ground\nT23 is ground\nT26 is ground\nT27 is ground\nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: split(T24, X53, X54), SCOPE: 3, CLAUSE: 3 | split(T24, X53, X54), SCOPE: 3, CLAUSE: 4 | split(T24, X53, X54), SCOPE: 3, CLAUSE: 5\n\nT24 is ground\nX53 is free; \nX54 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: split(T24, X53, X54), SCOPE: 3, CLAUSE: 3\n\nT24 is ground\nX53 is free; \nX54 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: split(T24, X53, X54), SCOPE: 3, CLAUSE: 4 | split(T24, X53, X54), SCOPE: 3, CLAUSE: 5\n\nT24 is ground\nX53 is free; \nX54 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: split0(T33, X88, X89)\n\nT33 is ground\nX88 is free; \nX89 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: split0(T33, X88, X89), SCOPE: 4, CLAUSE: 6\n\nT33 is ground\nX88 is free; \nX89 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: split(T24, X53, X54), SCOPE: 3, CLAUSE: 4\n\nT24 is ground\nX53 is free; \nX54 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: split(T24, X53, X54), SCOPE: 3, CLAUSE: 5\n\nT24 is ground\nX53 is free; \nX54 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: split1(T38, X113, X114)\n\nT38 is ground\nX113 is free; \nX114 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: split1(T38, X113, X114), SCOPE: 5, CLAUSE: 7\n\nT38 is ground\nX113 is free; \nX114 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: split2(T47, X139, X140)\n\nT47 is ground\nX139 is free; \nX140 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: split2(T47, X139, X140), SCOPE: 6, CLAUSE: 8\n\nT47 is ground\nX139 is free; \nX140 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: split(T59, X167, X168)\n\nT59 is ground\nX167 is free; \nX168 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: mergesort(.(T22, T26), X23)\n\nT22 is ground\nT26 is ground\nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(mergesort(.(T23, T27), X24), merge(T63, X24, T14))\n\nT23 is ground\nT27 is ground\nT63 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: mergesort(.(T23, T27), X24)\n\nT23 is ground\nT27 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: merge(T63, T68, T14)\n\nT63 is ground\nT68 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: merge(T63, T68, T14), SCOPE: 7, CLAUSE: 9 | merge(T63, T68, T14), SCOPE: 7, CLAUSE: 10 | merge(T63, T68, T14), SCOPE: 7, CLAUSE: 11 | merge(T63, T68, T14), SCOPE: 7, CLAUSE: 12\n\nT63 is ground\nT68 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: merge(T63, T68, T14), SCOPE: 7, CLAUSE: 9\n\nT63 is ground\nT68 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: merge(T63, T68, T14), SCOPE: 7, CLAUSE: 10 | merge(T63, T68, T14), SCOPE: 7, CLAUSE: 11 | merge(T63, T68, T14), SCOPE: 7, CLAUSE: 12\n\nT63 is ground\nT68 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: merge(T63, T68, T14), SCOPE: 7, CLAUSE: 10\n\nT63 is ground\nT68 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: merge(T63, T68, T14), SCOPE: 7, CLAUSE: 11 | merge(T63, T68, T14), SCOPE: 7, CLAUSE: 12\n\nT63 is ground\nT68 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: merge(T63, T68, T14), SCOPE: 7, CLAUSE: 11\n\nT63 is ground\nT68 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: merge(T63, T68, T14), SCOPE: 7, CLAUSE: 12\n\nT63 is ground\nT68 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(le(T103, T105), merge(T104, .(T105, T106), T108))\n\nT103 is ground\nT104 is ground\nT105 is ground\nT106 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: le(T103, T105)\n\nT103 is ground\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: merge(T104, .(T105, T106), T108)\n\nT104 is ground\nT105 is ground\nT106 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: le(T103, T105), SCOPE: 8, CLAUSE: 15 | le(T103, T105), SCOPE: 8, CLAUSE: 16 | le(T103, T105), SCOPE: 8, CLAUSE: 17\n\nT103 is ground\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: le(T103, T105), SCOPE: 8, CLAUSE: 15\n\nT103 is ground\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: le(T103, T105), SCOPE: 8, CLAUSE: 16 | le(T103, T105), SCOPE: 8, CLAUSE: 17\n\nT103 is ground\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: le(T121, T122)\n\nT121 is ground\nT122 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: le(T103, T105), SCOPE: 8, CLAUSE: 16\n\nT103 is ground\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: le(T103, T105), SCOPE: 8, CLAUSE: 17\n\nT103 is ground\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(gt(T144, T146), merge(.(T144, T145), T147, T149))\n\nT144 is ground\nT145 is ground\nT146 is ground\nT147 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: gt(T144, T146)\n\nT144 is ground\nT146 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: merge(.(T144, T145), T147, T149)\n\nT144 is ground\nT145 is ground\nT147 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: gt(T144, T146), SCOPE: 9, CLAUSE: 13 | gt(T144, T146), SCOPE: 9, CLAUSE: 14\n\nT144 is ground\nT146 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: gt(T144, T146), SCOPE: 9, CLAUSE: 13\n\nT144 is ground\nT146 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: gt(T144, T146), SCOPE: 9, CLAUSE: 14\n\nT144 is ground\nT146 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: gt(T162, T163)\n\nT162 is ground\nT163 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 83 [arrowhead = none ];
83 -> 2;

84 [label="EVAL with clause\nmergesort([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 84 [arrowhead = none ];
84 -> 3;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 85 [arrowhead = none ];
85 -> 4;

86 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 86 [arrowhead = none ];
86 -> 5;

87 [label="EVAL with clause\nmergesort(.(X7, []), .(X7, [])).\nand substitution\nX7 -> T4\nT1 -> .(T4, [])\nT2 -> .(T4, [])", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 87 [arrowhead = none ];
87 -> 8;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 88 [arrowhead = none ];
88 -> 9;

89 [label="BACKTRACK\nfor clause: mergesort(.(X, []), .(X, []))because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 89 [arrowhead = none ];
89 -> 6;

90 [label="BACKTRACK\nfor clause: mergesort(.(X, .(Y, L1)), L2) :- ','(split2(.(X, .(Y, L1)), L3, L4), ','(mergesort(L3, L5), ','(mergesort(L4, L6), merge(L5, L6, L2))))because of non-unification", fontsize=14, style = filled, fillcolor = white];
6 -> 90 [arrowhead = none ];
90 -> 7;

91 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 91 [arrowhead = none ];
91 -> 10;

92 [label="EVAL with clause\nmergesort(.(X17, .(X18, X19)), X20) :- ','(split2(.(X17, .(X18, X19)), X21, X22), ','(mergesort(X21, X23), ','(mergesort(X22, X24), merge(X23, X24, X20)))).\nand substitution\nX17 -> T10\nX18 -> T11\nX19 -> T12\nT1 -> .(T10, .(T11, T12))\nT2 -> T14\nX20 -> T14\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 92 [arrowhead = none ];
92 -> 12;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 93 [arrowhead = none ];
93 -> 13;

94 [label="BACKTRACK\nfor clause: mergesort(.(X, .(Y, L1)), L2) :- ','(split2(.(X, .(Y, L1)), L3, L4), ','(mergesort(L3, L5), ','(mergesort(L4, L6), merge(L5, L6, L2))))because of non-unification", fontsize=14, style = filled, fillcolor = white];
10 -> 94 [arrowhead = none ];
94 -> 11;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 95 [arrowhead = none ];
95 -> 14;

96 [label="ONLY EVAL with clause\nsplit2(.(X48, .(X49, X50)), .(X48, X51), .(X49, X52)) :- split(X50, X51, X52).\nand substitution\nT10 -> T22\nX48 -> T22\nT11 -> T23\nX49 -> T23\nT12 -> T24\nX50 -> T24\nX51 -> X53\nX21 -> .(T22, X53)\nX52 -> X54\nX22 -> .(T23, X54)", fontsize=14, style = filled, fillcolor = white];
14 -> 96 [arrowhead = none ];
96 -> 15;

97 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
15 -> 97 [arrowhead = none ];
97 -> 16;

98 [label="SPLIT 2\nnew knowledge:\nT26 is ground\nT27 is ground\nreplacements:\nX53 -> T26\nX54 -> T27", fontsize=14, style = filled, fillcolor = violet];
15 -> 98 [arrowhead = none ];
98 -> 17;

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 99 [arrowhead = none ];
99 -> 18;

100 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
17 -> 100 [arrowhead = none ];
100 -> 37;

101 [label="SPLIT 2\nnew knowledge:\nT63 is ground\nreplacements:\nX23 -> T63", fontsize=14, style = filled, fillcolor = violet];
17 -> 101 [arrowhead = none ];
101 -> 38;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 102 [arrowhead = none ];
102 -> 19;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 103 [arrowhead = none ];
103 -> 20;

104 [label="ONLY EVAL with clause\nsplit(X85, X86, X87) :- split0(X85, X86, X87).\nand substitution\nT24 -> T33\nX85 -> T33\nX53 -> X88\nX86 -> X88\nX54 -> X89\nX87 -> X89", fontsize=14, style = filled, fillcolor = white];
19 -> 104 [arrowhead = none ];
104 -> 21;

105 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 105 [arrowhead = none ];
105 -> 26;

106 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 106 [arrowhead = none ];
106 -> 27;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 107 [arrowhead = none ];
107 -> 22;

108 [label="EVAL with clause\nsplit0([], [], []).\nand substitution\nT33 -> []\nX88 -> []\nX89 -> []", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 108 [arrowhead = none ];
108 -> 23;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 109 [arrowhead = none ];
109 -> 24;

110 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 110 [arrowhead = none ];
110 -> 25;

111 [label="ONLY EVAL with clause\nsplit(X110, X111, X112) :- split1(X110, X111, X112).\nand substitution\nT24 -> T38\nX110 -> T38\nX53 -> X113\nX111 -> X113\nX54 -> X114\nX112 -> X114", fontsize=14, style = filled, fillcolor = white];
26 -> 111 [arrowhead = none ];
111 -> 28;

112 [label="ONLY EVAL with clause\nsplit(X136, X137, X138) :- split2(X136, X137, X138).\nand substitution\nT24 -> T47\nX136 -> T47\nX53 -> X139\nX137 -> X139\nX54 -> X140\nX138 -> X140", fontsize=14, style = filled, fillcolor = white];
27 -> 112 [arrowhead = none ];
112 -> 33;

113 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 113 [arrowhead = none ];
113 -> 29;

114 [label="EVAL with clause\nsplit1(.(X117, []), .(X117, []), []).\nand substitution\nX117 -> T41\nT38 -> .(T41, [])\nX113 -> .(T41, [])\nX114 -> []", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 114 [arrowhead = none ];
114 -> 30;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 115 [arrowhead = none ];
115 -> 31;

116 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 116 [arrowhead = none ];
116 -> 32;

117 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 117 [arrowhead = none ];
117 -> 34;

118 [label="EVAL with clause\nsplit2(.(X162, .(X163, X164)), .(X162, X165), .(X163, X166)) :- split(X164, X165, X166).\nand substitution\nX162 -> T57\nX163 -> T58\nX164 -> T59\nT47 -> .(T57, .(T58, T59))\nX165 -> X167\nX139 -> .(T57, X167)\nX166 -> X168\nX140 -> .(T58, X168)", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 118 [arrowhead = none ];
118 -> 35;

119 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 119 [arrowhead = none ];
119 -> 36;

120 [label="INSTANCE with matching:\nT24 -> T59\nX53 -> X167\nX54 -> X168", fontsize=14, style = filled, fillcolor = lightgrey];
35 -> 120 [arrowhead = none , style=dashed];
120 -> 16[style=dashed];

121 [label="INSTANCE with matching:\nT1 -> .(T22, T26)\nT2 -> X23", fontsize=14, style = filled, fillcolor = lightgrey];
37 -> 121 [arrowhead = none , style=dashed];
121 -> 1[style=dashed];

122 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
38 -> 122 [arrowhead = none ];
122 -> 39;

123 [label="SPLIT 2\nnew knowledge:\nT68 is ground\nreplacements:\nX24 -> T68", fontsize=14, style = filled, fillcolor = violet];
38 -> 123 [arrowhead = none ];
123 -> 40;

124 [label="INSTANCE with matching:\nT1 -> .(T23, T27)\nT2 -> X24", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 124 [arrowhead = none , style=dashed];
124 -> 1[style=dashed];

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 125 [arrowhead = none ];
125 -> 41;

126 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 126 [arrowhead = none ];
126 -> 42;

127 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 127 [arrowhead = none ];
127 -> 43;

128 [label="EVAL with clause\nmerge([], X193, X193).\nand substitution\nT63 -> []\nT68 -> T77\nX193 -> T77\nT14 -> T77", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 128 [arrowhead = none ];
128 -> 44;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 129 [arrowhead = none ];
129 -> 45;

130 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 130 [arrowhead = none ];
130 -> 47;

131 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 131 [arrowhead = none ];
131 -> 48;

132 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 132 [arrowhead = none ];
132 -> 46;

133 [label="EVAL with clause\nmerge(X199, [], X199).\nand substitution\nT63 -> T82\nX199 -> T82\nT68 -> []\nT14 -> T82", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 133 [arrowhead = none ];
133 -> 49;

134 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 134 [arrowhead = none ];
134 -> 50;

135 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 135 [arrowhead = none ];
135 -> 52;

136 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
48 -> 136 [arrowhead = none ];
136 -> 53;

137 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 137 [arrowhead = none ];
137 -> 51;

138 [label="EVAL with clause\nmerge(.(X221, X222), .(X223, X224), .(X221, X225)) :- ','(le(X221, X223), merge(X222, .(X223, X224), X225)).\nand substitution\nX221 -> T103\nX222 -> T104\nT63 -> .(T103, T104)\nX223 -> T105\nX224 -> T106\nT68 -> .(T105, T106)\nX225 -> T108\nT14 -> .(T103, T108)\nT107 -> T108", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 138 [arrowhead = none ];
138 -> 54;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 139 [arrowhead = none ];
139 -> 55;

140 [label="EVAL with clause\nmerge(.(X267, X268), .(X269, X270), .(X269, X271)) :- ','(gt(X267, X269), merge(.(X267, X268), X270, X271)).\nand substitution\nX267 -> T144\nX268 -> T145\nT63 -> .(T144, T145)\nX269 -> T146\nX270 -> T147\nT68 -> .(T146, T147)\nX271 -> T149\nT14 -> .(T146, T149)\nT148 -> T149", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 140 [arrowhead = none ];
140 -> 71;

141 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 141 [arrowhead = none ];
141 -> 72;

142 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
54 -> 142 [arrowhead = none ];
142 -> 56;

143 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
54 -> 143 [arrowhead = none ];
143 -> 57;

144 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
56 -> 144 [arrowhead = none ];
144 -> 58;

145 [label="INSTANCE with matching:\nT63 -> T104\nT68 -> .(T105, T106)\nT14 -> T108", fontsize=14, style = filled, fillcolor = lightgrey];
57 -> 145 [arrowhead = none , style=dashed];
145 -> 40[style=dashed];

146 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
58 -> 146 [arrowhead = none ];
146 -> 59;

147 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
58 -> 147 [arrowhead = none ];
147 -> 60;

148 [label="EVAL with clause\nle(s(X243), s(X244)) :- le(X243, X244).\nand substitution\nX243 -> T121\nT103 -> s(T121)\nX244 -> T122\nT105 -> s(T122)", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 148 [arrowhead = none ];
148 -> 61;

149 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 149 [arrowhead = none ];
149 -> 62;

150 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 150 [arrowhead = none ];
150 -> 63;

151 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 151 [arrowhead = none ];
151 -> 64;

152 [label="INSTANCE with matching:\nT103 -> T121\nT105 -> T122", fontsize=14, style = filled, fillcolor = lightgrey];
61 -> 152 [arrowhead = none , style=dashed];
152 -> 56[style=dashed];

153 [label="EVAL with clause\nle(0, s(X253)).\nand substitution\nT103 -> 0\nX253 -> T129\nT105 -> s(T129)", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 153 [arrowhead = none ];
153 -> 65;

154 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 154 [arrowhead = none ];
154 -> 66;

155 [label="EVAL with clause\nle(0, 0).\nand substitution\nT103 -> 0\nT105 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 155 [arrowhead = none ];
155 -> 68;

156 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 156 [arrowhead = none ];
156 -> 69;

157 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 157 [arrowhead = none ];
157 -> 67;

158 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 158 [arrowhead = none ];
158 -> 70;

159 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
71 -> 159 [arrowhead = none ];
159 -> 73;

160 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
71 -> 160 [arrowhead = none ];
160 -> 74;

161 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
73 -> 161 [arrowhead = none ];
161 -> 75;

162 [label="INSTANCE with matching:\nT63 -> .(T144, T145)\nT68 -> T147\nT14 -> T149", fontsize=14, style = filled, fillcolor = lightgrey];
74 -> 162 [arrowhead = none , style=dashed];
162 -> 40[style=dashed];

163 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
75 -> 163 [arrowhead = none ];
163 -> 76;

164 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
75 -> 164 [arrowhead = none ];
164 -> 77;

165 [label="EVAL with clause\ngt(s(X289), s(X290)) :- gt(X289, X290).\nand substitution\nX289 -> T162\nT144 -> s(T162)\nX290 -> T163\nT146 -> s(T163)", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 165 [arrowhead = none ];
165 -> 78;

166 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 166 [arrowhead = none ];
166 -> 79;

167 [label="EVAL with clause\ngt(s(X297), 0).\nand substitution\nX297 -> T168\nT144 -> s(T168)\nT146 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
77 -> 167 [arrowhead = none ];
167 -> 80;

168 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
77 -> 168 [arrowhead = none ];
168 -> 81;

169 [label="INSTANCE with matching:\nT144 -> T162\nT146 -> T163", fontsize=14, style = filled, fillcolor = lightgrey];
78 -> 169 [arrowhead = none , style=dashed];
169 -> 73[style=dashed];

170 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
80 -> 170 [arrowhead = none ];
170 -> 82;

}

(2) Obligation:

Clauses:

split16([], [], []).
split16(.(T41, []), .(T41, []), []).
split16(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) :- split16(T59, X167, X168).
merge40([], T77, T77).
merge40(T82, [], T82).
merge40(.(T103, T104), .(T105, T106), .(T103, T108)) :- le56(T103, T105).
merge40(.(T103, T104), .(T105, T106), .(T103, T108)) :- ','(le56(T103, T105), merge40(T104, .(T105, T106), T108)).
merge40(.(T144, T145), .(T146, T147), .(T146, T149)) :- gt73(T144, T146).
merge40(.(T144, T145), .(T146, T147), .(T146, T149)) :- ','(gt73(T144, T146), merge40(.(T144, T145), T147, T149)).
le56(s(T121), s(T122)) :- le56(T121, T122).
le56(0, s(T129)).
le56(0, 0).
gt73(s(T162), s(T163)) :- gt73(T162, T163).
gt73(s(T168), 0).
mergesort1([], []).
mergesort1(.(T4, []), .(T4, [])).
mergesort1(.(T22, .(T23, T24)), T14) :- split16(T24, X53, X54).
mergesort1(.(T22, .(T23, T24)), T14) :- ','(split16(T24, T26, T27), mergesort1(.(T22, T26), X23)).
mergesort1(.(T22, .(T23, T24)), T14) :- ','(split16(T24, T26, T27), ','(mergesort1(.(T22, T26), T63), mergesort1(.(T23, T27), X24))).
mergesort1(.(T22, .(T23, T24)), T14) :- ','(split16(T24, T26, T27), ','(mergesort1(.(T22, T26), T63), ','(mergesort1(.(T23, T27), T68), merge40(T63, T68, T14)))).

Queries:

mergesort1(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
mergesort1_in: (b,f)
split16_in: (b,f,f)
merge40_in: (f,f,f)
le56_in: (f,f)
gt73_in: (f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U8_ga(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
split16_in_gaa([], [], []) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, []), .(T41, []), []) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_gaa(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
U1_gaa(T57, T58, T59, X167, X168, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, T14, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U9_ga(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U10_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T22, T26), X23)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U12_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T23, T27), X24)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_ga(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_ga(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_ga(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
merge40_in_aaa([], T77, T77) → merge40_out_aaa([], T77, T77)
merge40_in_aaa(T82, [], T82) → merge40_out_aaa(T82, [], T82)
merge40_in_aaa(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_aaa(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_aaa(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
merge40_in_aaa(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_aaa(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_aaa(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U5_aaa(T144, T145, T146, T147, T149, merge40_out_aaa(.(T144, T145), T147, T149)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U3_aaa(T103, T104, T105, T106, T108, merge40_out_aaa(T104, .(T105, T106), T108)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U14_ga(T22, T23, T24, T14, merge40_out_aaa(T63, T68, T14)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U8_ga(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
split16_in_gaa([], [], []) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, []), .(T41, []), []) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_gaa(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
U1_gaa(T57, T58, T59, X167, X168, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, T14, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U9_ga(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U10_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T22, T26), X23)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U12_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T23, T27), X24)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_ga(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_ga(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_ga(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
merge40_in_aaa([], T77, T77) → merge40_out_aaa([], T77, T77)
merge40_in_aaa(T82, [], T82) → merge40_out_aaa(T82, [], T82)
merge40_in_aaa(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_aaa(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_aaa(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
merge40_in_aaa(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_aaa(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_aaa(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U5_aaa(T144, T145, T146, T147, T149, merge40_out_aaa(.(T144, T145), T147, T149)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U3_aaa(T103, T104, T105, T106, T108, merge40_out_aaa(T104, .(T105, T106), T108)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U14_ga(T22, T23, T24, T14, merge40_out_aaa(T63, T68, T14)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

MERGESORT1_IN_GA(.(T22, .(T23, T24)), T14) → U8_GA(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
MERGESORT1_IN_GA(.(T22, .(T23, T24)), T14) → SPLIT16_IN_GAA(T24, X53, X54)
SPLIT16_IN_GAA(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_GAA(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
SPLIT16_IN_GAA(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → SPLIT16_IN_GAA(T59, X167, X168)
MERGESORT1_IN_GA(.(T22, .(T23, T24)), T14) → U9_GA(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_GA(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_GA(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U9_GA(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → MERGESORT1_IN_GA(.(T22, T26), X23)
U9_GA(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_GA(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_GA(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_GA(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U11_GA(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → MERGESORT1_IN_GA(.(T23, T27), X24)
U11_GA(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_GA(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_GA(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_GA(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
U13_GA(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → MERGE40_IN_AAA(T63, T68, T14)
MERGE40_IN_AAA(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_AAA(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
MERGE40_IN_AAA(.(T103, T104), .(T105, T106), .(T103, T108)) → LE56_IN_AA(T103, T105)
LE56_IN_AA(s(T121), s(T122)) → U6_AA(T121, T122, le56_in_aa(T121, T122))
LE56_IN_AA(s(T121), s(T122)) → LE56_IN_AA(T121, T122)
U2_AAA(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_AAA(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
U2_AAA(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → MERGE40_IN_AAA(T104, .(T105, T106), T108)
MERGE40_IN_AAA(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_AAA(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
MERGE40_IN_AAA(.(T144, T145), .(T146, T147), .(T146, T149)) → GT73_IN_AA(T144, T146)
GT73_IN_AA(s(T162), s(T163)) → U7_AA(T162, T163, gt73_in_aa(T162, T163))
GT73_IN_AA(s(T162), s(T163)) → GT73_IN_AA(T162, T163)
U4_AAA(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_AAA(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U4_AAA(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → MERGE40_IN_AAA(.(T144, T145), T147, T149)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U8_ga(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
split16_in_gaa([], [], []) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, []), .(T41, []), []) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_gaa(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
U1_gaa(T57, T58, T59, X167, X168, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, T14, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U9_ga(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U10_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T22, T26), X23)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U12_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T23, T27), X24)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_ga(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_ga(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_ga(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
merge40_in_aaa([], T77, T77) → merge40_out_aaa([], T77, T77)
merge40_in_aaa(T82, [], T82) → merge40_out_aaa(T82, [], T82)
merge40_in_aaa(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_aaa(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_aaa(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
merge40_in_aaa(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_aaa(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_aaa(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U5_aaa(T144, T145, T146, T147, T149, merge40_out_aaa(.(T144, T145), T147, T149)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U3_aaa(T103, T104, T105, T106, T108, merge40_out_aaa(T104, .(T105, T106), T108)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U14_ga(T22, T23, T24, T14, merge40_out_aaa(T63, T68, T14)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2) = mergesort1_in_ga(x1)
[] = []
mergesort1_out_ga(x1, x2) = mergesort1_out_ga
.(x1, x2) = .(x1, x2)
U8_ga(x1, x2, x3, x4, x5) = U8_ga(x5)
split16_in_gaa(x1, x2, x3) = split16_in_gaa(x1)
split16_out_gaa(x1, x2, x3) = split16_out_gaa(x2, x3)
U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x2, x6)
U9_ga(x1, x2, x3, x4, x5) = U9_ga(x1, x2, x5)
U10_ga(x1, x2, x3, x4, x5) = U10_ga(x5)
U11_ga(x1, x2, x3, x4, x5, x6) = U11_ga(x2, x5, x6)
U12_ga(x1, x2, x3, x4, x5) = U12_ga(x5)
U13_ga(x1, x2, x3, x4, x5, x6) = U13_ga(x6)
U14_ga(x1, x2, x3, x4, x5) = U14_ga(x5)
merge40_in_aaa(x1, x2, x3) = merge40_in_aaa
merge40_out_aaa(x1, x2, x3) = merge40_out_aaa
U2_aaa(x1, x2, x3, x4, x5, x6) = U2_aaa(x6)
le56_in_aa(x1, x2) = le56_in_aa
U6_aa(x1, x2, x3) = U6_aa(x3)
le56_out_aa(x1, x2) = le56_out_aa(x1)
U3_aaa(x1, x2, x3, x4, x5, x6) = U3_aaa(x6)
U4_aaa(x1, x2, x3, x4, x5, x6) = U4_aaa(x6)
gt73_in_aa(x1, x2) = gt73_in_aa
U7_aa(x1, x2, x3) = U7_aa(x3)
gt73_out_aa(x1, x2) = gt73_out_aa(x2)
U5_aaa(x1, x2, x3, x4, x5, x6) = U5_aaa(x6)
s(x1) = s(x1)
MERGESORT1_IN_GA(x1, x2) = MERGESORT1_IN_GA(x1)
U8_GA(x1, x2, x3, x4, x5) = U8_GA(x5)
SPLIT16_IN_GAA(x1, x2, x3) = SPLIT16_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x1, x2, x6)
U9_GA(x1, x2, x3, x4, x5) = U9_GA(x1, x2, x5)
U10_GA(x1, x2, x3, x4, x5) = U10_GA(x5)
U11_GA(x1, x2, x3, x4, x5, x6) = U11_GA(x2, x5, x6)
U12_GA(x1, x2, x3, x4, x5) = U12_GA(x5)
U13_GA(x1, x2, x3, x4, x5, x6) = U13_GA(x6)
U14_GA(x1, x2, x3, x4, x5) = U14_GA(x5)
MERGE40_IN_AAA(x1, x2, x3) = MERGE40_IN_AAA
U2_AAA(x1, x2, x3, x4, x5, x6) = U2_AAA(x6)
LE56_IN_AA(x1, x2) = LE56_IN_AA
U6_AA(x1, x2, x3) = U6_AA(x3)
U3_AAA(x1, x2, x3, x4, x5, x6) = U3_AAA(x6)
U4_AAA(x1, x2, x3, x4, x5, x6) = U4_AAA(x6)
GT73_IN_AA(x1, x2) = GT73_IN_AA
U7_AA(x1, x2, x3) = U7_AA(x3)
U5_AAA(x1, x2, x3, x4, x5, x6) = U5_AAA(x6)

We have to consider all (P,R,Pi)-chains

(6) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MERGESORT1_IN_GA(.(T22, .(T23, T24)), T14) → U8_GA(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
MERGESORT1_IN_GA(.(T22, .(T23, T24)), T14) → SPLIT16_IN_GAA(T24, X53, X54)
SPLIT16_IN_GAA(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_GAA(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
SPLIT16_IN_GAA(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → SPLIT16_IN_GAA(T59, X167, X168)
MERGESORT1_IN_GA(.(T22, .(T23, T24)), T14) → U9_GA(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_GA(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_GA(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U9_GA(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → MERGESORT1_IN_GA(.(T22, T26), X23)
U9_GA(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_GA(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_GA(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_GA(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U11_GA(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → MERGESORT1_IN_GA(.(T23, T27), X24)
U11_GA(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_GA(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_GA(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_GA(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
U13_GA(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → MERGE40_IN_AAA(T63, T68, T14)
MERGE40_IN_AAA(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_AAA(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
MERGE40_IN_AAA(.(T103, T104), .(T105, T106), .(T103, T108)) → LE56_IN_AA(T103, T105)
LE56_IN_AA(s(T121), s(T122)) → U6_AA(T121, T122, le56_in_aa(T121, T122))
LE56_IN_AA(s(T121), s(T122)) → LE56_IN_AA(T121, T122)
U2_AAA(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_AAA(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
U2_AAA(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → MERGE40_IN_AAA(T104, .(T105, T106), T108)
MERGE40_IN_AAA(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_AAA(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
MERGE40_IN_AAA(.(T144, T145), .(T146, T147), .(T146, T149)) → GT73_IN_AA(T144, T146)
GT73_IN_AA(s(T162), s(T163)) → U7_AA(T162, T163, gt73_in_aa(T162, T163))
GT73_IN_AA(s(T162), s(T163)) → GT73_IN_AA(T162, T163)
U4_AAA(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_AAA(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U4_AAA(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → MERGE40_IN_AAA(.(T144, T145), T147, T149)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U8_ga(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
split16_in_gaa([], [], []) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, []), .(T41, []), []) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_gaa(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
U1_gaa(T57, T58, T59, X167, X168, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, T14, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U9_ga(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U10_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T22, T26), X23)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U12_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T23, T27), X24)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_ga(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_ga(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_ga(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
merge40_in_aaa([], T77, T77) → merge40_out_aaa([], T77, T77)
merge40_in_aaa(T82, [], T82) → merge40_out_aaa(T82, [], T82)
merge40_in_aaa(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_aaa(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_aaa(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
merge40_in_aaa(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_aaa(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_aaa(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U5_aaa(T144, T145, T146, T147, T149, merge40_out_aaa(.(T144, T145), T147, T149)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U3_aaa(T103, T104, T105, T106, T108, merge40_out_aaa(T104, .(T105, T106), T108)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U14_ga(T22, T23, T24, T14, merge40_out_aaa(T63, T68, T14)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 14 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

GT73_IN_AA(s(T162), s(T163)) → GT73_IN_AA(T162, T163)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U8_ga(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
split16_in_gaa([], [], []) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, []), .(T41, []), []) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_gaa(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
U1_gaa(T57, T58, T59, X167, X168, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, T14, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U9_ga(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U10_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T22, T26), X23)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U12_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T23, T27), X24)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_ga(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_ga(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_ga(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
merge40_in_aaa([], T77, T77) → merge40_out_aaa([], T77, T77)
merge40_in_aaa(T82, [], T82) → merge40_out_aaa(T82, [], T82)
merge40_in_aaa(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_aaa(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_aaa(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
merge40_in_aaa(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_aaa(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_aaa(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U5_aaa(T144, T145, T146, T147, T149, merge40_out_aaa(.(T144, T145), T147, T149)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U3_aaa(T103, T104, T105, T106, T108, merge40_out_aaa(T104, .(T105, T106), T108)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U14_ga(T22, T23, T24, T14, merge40_out_aaa(T63, T68, T14)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)

(10) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(11) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

GT73_IN_AA(s(T162), s(T163)) → GT73_IN_AA(T162, T163)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
GT73_IN_AA(x1, x2) = GT73_IN_AA

We have to consider all (P,R,Pi)-chains

(12) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(13) Obligation:

Q DP problem:
The TRS P consists of the following rules:

GT73_IN_AA → GT73_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(14) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = GT73_IN_AA evaluates to t =GT73_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from GT73_IN_AA to GT73_IN_AA.

(15) NO

(16) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LE56_IN_AA(s(T121), s(T122)) → LE56_IN_AA(T121, T122)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U8_ga(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
split16_in_gaa([], [], []) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, []), .(T41, []), []) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_gaa(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
U1_gaa(T57, T58, T59, X167, X168, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, T14, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U9_ga(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U10_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T22, T26), X23)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U12_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T23, T27), X24)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_ga(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_ga(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_ga(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
merge40_in_aaa([], T77, T77) → merge40_out_aaa([], T77, T77)
merge40_in_aaa(T82, [], T82) → merge40_out_aaa(T82, [], T82)
merge40_in_aaa(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_aaa(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_aaa(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
merge40_in_aaa(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_aaa(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_aaa(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U5_aaa(T144, T145, T146, T147, T149, merge40_out_aaa(.(T144, T145), T147, T149)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U3_aaa(T103, T104, T105, T106, T108, merge40_out_aaa(T104, .(T105, T106), T108)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U14_ga(T22, T23, T24, T14, merge40_out_aaa(T63, T68, T14)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)

(17) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(18) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LE56_IN_AA(s(T121), s(T122)) → LE56_IN_AA(T121, T122)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
LE56_IN_AA(x1, x2) = LE56_IN_AA

We have to consider all (P,R,Pi)-chains

(19) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LE56_IN_AA → LE56_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(21) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = LE56_IN_AA evaluates to t =LE56_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LE56_IN_AA to LE56_IN_AA.

(22) NO

(23) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

U2_AAA(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → MERGE40_IN_AAA(T104, .(T105, T106), T108)
MERGE40_IN_AAA(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_AAA(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
MERGE40_IN_AAA(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_AAA(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
U4_AAA(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → MERGE40_IN_AAA(.(T144, T145), T147, T149)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U8_ga(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
split16_in_gaa([], [], []) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, []), .(T41, []), []) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_gaa(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
U1_gaa(T57, T58, T59, X167, X168, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, T14, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U9_ga(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U10_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T22, T26), X23)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U12_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T23, T27), X24)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_ga(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_ga(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_ga(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
merge40_in_aaa([], T77, T77) → merge40_out_aaa([], T77, T77)
merge40_in_aaa(T82, [], T82) → merge40_out_aaa(T82, [], T82)
merge40_in_aaa(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_aaa(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_aaa(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
merge40_in_aaa(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_aaa(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_aaa(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U5_aaa(T144, T145, T146, T147, T149, merge40_out_aaa(.(T144, T145), T147, T149)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U3_aaa(T103, T104, T105, T106, T108, merge40_out_aaa(T104, .(T105, T106), T108)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U14_ga(T22, T23, T24, T14, merge40_out_aaa(T63, T68, T14)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)

(24) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(25) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

U2_AAA(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → MERGE40_IN_AAA(T104, .(T105, T106), T108)
MERGE40_IN_AAA(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_AAA(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
MERGE40_IN_AAA(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_AAA(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
U4_AAA(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → MERGE40_IN_AAA(.(T144, T145), T147, T149)

The TRS R consists of the following rules:

le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
le56_in_aa(x1, x2) = le56_in_aa
U6_aa(x1, x2, x3) = U6_aa(x3)
le56_out_aa(x1, x2) = le56_out_aa(x1)
gt73_in_aa(x1, x2) = gt73_in_aa
U7_aa(x1, x2, x3) = U7_aa(x3)
gt73_out_aa(x1, x2) = gt73_out_aa(x2)
s(x1) = s(x1)
MERGE40_IN_AAA(x1, x2, x3) = MERGE40_IN_AAA
U2_AAA(x1, x2, x3, x4, x5, x6) = U2_AAA(x6)
U4_AAA(x1, x2, x3, x4, x5, x6) = U4_AAA(x6)

We have to consider all (P,R,Pi)-chains

(26) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(27) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U2_AAA(le56_out_aa(T103)) → MERGE40_IN_AAA
MERGE40_IN_AAA → U2_AAA(le56_in_aa)
MERGE40_IN_AAA → U4_AAA(gt73_in_aa)
U4_AAA(gt73_out_aa(T146)) → MERGE40_IN_AAA

The TRS R consists of the following rules:

le56_in_aa → U6_aa(le56_in_aa)
le56_in_aa → le56_out_aa(0)
gt73_in_aa → U7_aa(gt73_in_aa)
gt73_in_aa → gt73_out_aa(0)
U6_aa(le56_out_aa(T121)) → le56_out_aa(s(T121))
U7_aa(gt73_out_aa(T163)) → gt73_out_aa(s(T163))

The set Q consists of the following terms:

le56_in_aa
gt73_in_aa
U6_aa(x0)
U7_aa(x0)

We have to consider all (P,Q,R)-chains.

(28) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MERGE40_IN_AAA → U2_AAA(le56_in_aa) at position [0] we obtained the following new rules [LPAR04]:

MERGE40_IN_AAA → U2_AAA(U6_aa(le56_in_aa))
MERGE40_IN_AAA → U2_AAA(le56_out_aa(0))

(29) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U2_AAA(le56_out_aa(T103)) → MERGE40_IN_AAA
MERGE40_IN_AAA → U4_AAA(gt73_in_aa)
U4_AAA(gt73_out_aa(T146)) → MERGE40_IN_AAA
MERGE40_IN_AAA → U2_AAA(U6_aa(le56_in_aa))
MERGE40_IN_AAA → U2_AAA(le56_out_aa(0))

The TRS R consists of the following rules:

le56_in_aa → U6_aa(le56_in_aa)
le56_in_aa → le56_out_aa(0)
gt73_in_aa → U7_aa(gt73_in_aa)
gt73_in_aa → gt73_out_aa(0)
U6_aa(le56_out_aa(T121)) → le56_out_aa(s(T121))
U7_aa(gt73_out_aa(T163)) → gt73_out_aa(s(T163))

The set Q consists of the following terms:

le56_in_aa
gt73_in_aa
U6_aa(x0)
U7_aa(x0)

We have to consider all (P,Q,R)-chains.

(30) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MERGE40_IN_AAA → U4_AAA(gt73_in_aa) at position [0] we obtained the following new rules [LPAR04]:

MERGE40_IN_AAA → U4_AAA(U7_aa(gt73_in_aa))
MERGE40_IN_AAA → U4_AAA(gt73_out_aa(0))

(31) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U2_AAA(le56_out_aa(T103)) → MERGE40_IN_AAA
U4_AAA(gt73_out_aa(T146)) → MERGE40_IN_AAA
MERGE40_IN_AAA → U2_AAA(U6_aa(le56_in_aa))
MERGE40_IN_AAA → U2_AAA(le56_out_aa(0))
MERGE40_IN_AAA → U4_AAA(U7_aa(gt73_in_aa))
MERGE40_IN_AAA → U4_AAA(gt73_out_aa(0))

The TRS R consists of the following rules:

le56_in_aa → U6_aa(le56_in_aa)
le56_in_aa → le56_out_aa(0)
gt73_in_aa → U7_aa(gt73_in_aa)
gt73_in_aa → gt73_out_aa(0)
U6_aa(le56_out_aa(T121)) → le56_out_aa(s(T121))
U7_aa(gt73_out_aa(T163)) → gt73_out_aa(s(T163))

The set Q consists of the following terms:

le56_in_aa
gt73_in_aa
U6_aa(x0)
U7_aa(x0)

We have to consider all (P,Q,R)-chains.

(32) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = MERGE40_IN_AAA evaluates to t =MERGE40_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

MERGE40_IN_AAA → U2_AAA(le56_out_aa(0))
with rule MERGE40_IN_AAA → U2_AAA(le56_out_aa(0)) at position [] and matcher [ ]

U2_AAA(le56_out_aa(0)) → MERGE40_IN_AAA
with rule U2_AAA(le56_out_aa(T103)) → MERGE40_IN_AAA

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

(33) NO

(34) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SPLIT16_IN_GAA(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → SPLIT16_IN_GAA(T59, X167, X168)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U8_ga(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
split16_in_gaa([], [], []) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, []), .(T41, []), []) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_gaa(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
U1_gaa(T57, T58, T59, X167, X168, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, T14, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U9_ga(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U10_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T22, T26), X23)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U12_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T23, T27), X24)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_ga(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_ga(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_ga(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
merge40_in_aaa([], T77, T77) → merge40_out_aaa([], T77, T77)
merge40_in_aaa(T82, [], T82) → merge40_out_aaa(T82, [], T82)
merge40_in_aaa(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_aaa(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_aaa(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
merge40_in_aaa(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_aaa(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_aaa(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U5_aaa(T144, T145, T146, T147, T149, merge40_out_aaa(.(T144, T145), T147, T149)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U3_aaa(T103, T104, T105, T106, T108, merge40_out_aaa(T104, .(T105, T106), T108)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U14_ga(T22, T23, T24, T14, merge40_out_aaa(T63, T68, T14)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)

(35) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(36) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SPLIT16_IN_GAA(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → SPLIT16_IN_GAA(T59, X167, X168)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
SPLIT16_IN_GAA(x1, x2, x3) = SPLIT16_IN_GAA(x1)

We have to consider all (P,R,Pi)-chains

(37) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(38) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SPLIT16_IN_GAA(.(T57, .(T58, T59))) → SPLIT16_IN_GAA(T59)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(39) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

SPLIT16_IN_GAA(.(T57, .(T58, T59))) → SPLIT16_IN_GAA(T59)
The graph contains the following edges 1 > 1

(40) YES

(41) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MERGESORT1_IN_GA(.(T22, .(T23, T24)), T14) → U9_GA(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_GA(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → MERGESORT1_IN_GA(.(T22, T26), X23)
U9_GA(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_GA(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_GA(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → MERGESORT1_IN_GA(.(T23, T27), X24)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U8_ga(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
split16_in_gaa([], [], []) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, []), .(T41, []), []) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_gaa(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
U1_gaa(T57, T58, T59, X167, X168, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, T14, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U9_ga(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U10_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T22, T26), X23)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U12_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T23, T27), X24)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_ga(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_ga(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_ga(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
merge40_in_aaa([], T77, T77) → merge40_out_aaa([], T77, T77)
merge40_in_aaa(T82, [], T82) → merge40_out_aaa(T82, [], T82)
merge40_in_aaa(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_aaa(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_aaa(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
merge40_in_aaa(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_aaa(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_aaa(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U5_aaa(T144, T145, T146, T147, T149, merge40_out_aaa(.(T144, T145), T147, T149)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U3_aaa(T103, T104, T105, T106, T108, merge40_out_aaa(T104, .(T105, T106), T108)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U14_ga(T22, T23, T24, T14, merge40_out_aaa(T63, T68, T14)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)

(42) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(43) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MERGESORT1_IN_GA(.(T22, .(T23, T24)), T14) → U9_GA(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_GA(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → MERGESORT1_IN_GA(.(T22, T26), X23)
U9_GA(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_GA(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_GA(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → MERGESORT1_IN_GA(.(T23, T27), X24)

The TRS R consists of the following rules:

split16_in_gaa([], [], []) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, []), .(T41, []), []) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_gaa(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U8_ga(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U9_ga(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U1_gaa(T57, T58, T59, X167, X168, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, T14, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U10_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T22, T26), X23)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_ga(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U12_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T23, T27), X24)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U13_ga(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_ga(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
U14_ga(T22, T23, T24, T14, merge40_out_aaa(T63, T68, T14)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
merge40_in_aaa([], T77, T77) → merge40_out_aaa([], T77, T77)
merge40_in_aaa(T82, [], T82) → merge40_out_aaa(T82, [], T82)
merge40_in_aaa(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_aaa(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
merge40_in_aaa(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_aaa(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_aaa(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_aaa(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
U3_aaa(T103, T104, T105, T106, T108, merge40_out_aaa(T104, .(T105, T106), T108)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U5_aaa(T144, T145, T146, T147, T149, merge40_out_aaa(.(T144, T145), T147, T149)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))

(44) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(45) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MERGESORT1_IN_GA(.(T22, .(T23, T24))) → U9_GA(T22, T23, split16_in_gaa(T24))
U9_GA(T22, T23, split16_out_gaa(T26, T27)) → MERGESORT1_IN_GA(.(T22, T26))
U9_GA(T22, T23, split16_out_gaa(T26, T27)) → U11_GA(T23, T27, mergesort1_in_ga(.(T22, T26)))
U11_GA(T23, T27, mergesort1_out_ga) → MERGESORT1_IN_GA(.(T23, T27))

The TRS R consists of the following rules:

split16_in_gaa([]) → split16_out_gaa([], [])
split16_in_gaa(.(T41, [])) → split16_out_gaa(.(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59))) → U1_gaa(T57, T58, split16_in_gaa(T59))
mergesort1_in_ga(.(T4, [])) → mergesort1_out_ga
mergesort1_in_ga(.(T22, .(T23, T24))) → U8_ga(split16_in_gaa(T24))
mergesort1_in_ga(.(T22, .(T23, T24))) → U9_ga(T22, T23, split16_in_gaa(T24))
U1_gaa(T57, T58, split16_out_gaa(X167, X168)) → split16_out_gaa(.(T57, X167), .(T58, X168))
U8_ga(split16_out_gaa(X53, X54)) → mergesort1_out_ga
U9_ga(T22, T23, split16_out_gaa(T26, T27)) → U10_ga(mergesort1_in_ga(.(T22, T26)))
U9_ga(T22, T23, split16_out_gaa(T26, T27)) → U11_ga(T23, T27, mergesort1_in_ga(.(T22, T26)))
U10_ga(mergesort1_out_ga) → mergesort1_out_ga
U11_ga(T23, T27, mergesort1_out_ga) → U12_ga(mergesort1_in_ga(.(T23, T27)))
U11_ga(T23, T27, mergesort1_out_ga) → U13_ga(mergesort1_in_ga(.(T23, T27)))
U12_ga(mergesort1_out_ga) → mergesort1_out_ga
U13_ga(mergesort1_out_ga) → U14_ga(merge40_in_aaa)
U14_ga(merge40_out_aaa) → mergesort1_out_ga
merge40_in_aaa → merge40_out_aaa
merge40_in_aaa → U2_aaa(le56_in_aa)
merge40_in_aaa → U4_aaa(gt73_in_aa)
U2_aaa(le56_out_aa(T103)) → merge40_out_aaa
U2_aaa(le56_out_aa(T103)) → U3_aaa(merge40_in_aaa)
U4_aaa(gt73_out_aa(T146)) → merge40_out_aaa
U4_aaa(gt73_out_aa(T146)) → U5_aaa(merge40_in_aaa)
le56_in_aa → U6_aa(le56_in_aa)
le56_in_aa → le56_out_aa(0)
U3_aaa(merge40_out_aaa) → merge40_out_aaa
gt73_in_aa → U7_aa(gt73_in_aa)
gt73_in_aa → gt73_out_aa(0)
U5_aaa(merge40_out_aaa) → merge40_out_aaa
U6_aa(le56_out_aa(T121)) → le56_out_aa(s(T121))
U7_aa(gt73_out_aa(T163)) → gt73_out_aa(s(T163))

The set Q consists of the following terms:

split16_in_gaa(x0)
mergesort1_in_ga(x0)
U1_gaa(x0, x1, x2)
U8_ga(x0)
U9_ga(x0, x1, x2)
U10_ga(x0)
U11_ga(x0, x1, x2)
U12_ga(x0)
U13_ga(x0)
U14_ga(x0)
merge40_in_aaa
U2_aaa(x0)
U4_aaa(x0)
le56_in_aa
U3_aaa(x0)
gt73_in_aa
U5_aaa(x0)
U6_aa(x0)
U7_aa(x0)

We have to consider all (P,Q,R)-chains.

(46) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

MERGESORT1_IN_GA(.(T22, .(T23, T24))) → U9_GA(T22, T23, split16_in_gaa(T24))
U9_GA(T22, T23, split16_out_gaa(T26, T27)) → MERGESORT1_IN_GA(.(T22, T26))
U9_GA(T22, T23, split16_out_gaa(T26, T27)) → U11_GA(T23, T27, mergesort1_in_ga(.(T22, T26)))
U11_GA(T23, T27, mergesort1_out_ga) → MERGESORT1_IN_GA(.(T23, T27))

Strictly oriented rules of the TRS R:

split16_in_gaa([]) → split16_out_gaa([], [])
split16_in_gaa(.(T41, [])) → split16_out_gaa(.(T41, []), [])
mergesort1_in_ga(.(T4, [])) → mergesort1_out_ga
mergesort1_in_ga(.(T22, .(T23, T24))) → U8_ga(split16_in_gaa(T24))
mergesort1_in_ga(.(T22, .(T23, T24))) → U9_ga(T22, T23, split16_in_gaa(T24))
U8_ga(split16_out_gaa(X53, X54)) → mergesort1_out_ga
U9_ga(T22, T23, split16_out_gaa(T26, T27)) → U10_ga(mergesort1_in_ga(.(T22, T26)))
U9_ga(T22, T23, split16_out_gaa(T26, T27)) → U11_ga(T23, T27, mergesort1_in_ga(.(T22, T26)))
U10_ga(mergesort1_out_ga) → mergesort1_out_ga
U11_ga(T23, T27, mergesort1_out_ga) → U12_ga(mergesort1_in_ga(.(T23, T27)))
U11_ga(T23, T27, mergesort1_out_ga) → U13_ga(mergesort1_in_ga(.(T23, T27)))
U12_ga(mergesort1_out_ga) → mergesort1_out_ga
U13_ga(mergesort1_out_ga) → U14_ga(merge40_in_aaa)
U14_ga(merge40_out_aaa) → mergesort1_out_ga
merge40_in_aaa → merge40_out_aaa
U2_aaa(le56_out_aa(T103)) → merge40_out_aaa
U4_aaa(gt73_out_aa(T146)) → merge40_out_aaa

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 8 + x₁ + x₂
POL(0) = 0
POL(MERGESORT1_IN_GA(x₁)) = x₁
POL(U10_ga(x₁)) = 1 + x₁
POL(U11_GA(x₁, x₂, x₃)) = 2 + x₁ + x₂ + x₃
POL(U11_ga(x₁, x₂, x₃)) = 5 + x₁ + x₂ + x₃
POL(U12_ga(x₁)) = 1 + x₁
POL(U13_ga(x₁)) = 3 + x₁
POL(U14_ga(x₁)) = 8 + x₁
POL(U1_gaa(x₁, x₂, x₃)) = 16 + x₁ + x₂ + x₃
POL(U2_aaa(x₁)) = x₁
POL(U3_aaa(x₁)) = x₁
POL(U4_aaa(x₁)) = 1 + x₁
POL(U5_aaa(x₁)) = x₁
POL(U6_aa(x₁)) = x₁
POL(U7_aa(x₁)) = x₁
POL(U8_ga(x₁)) = 8 + x₁
POL(U9_GA(x₁, x₂, x₃)) = 11 + x₁ + x₂ + x₃
POL(U9_ga(x₁, x₂, x₃)) = 14 + x₁ + x₂ + x₃
POL([]) = 0
POL(gt73_in_aa) = 0
POL(gt73_out_aa(x₁)) = x₁
POL(le56_in_aa) = 1
POL(le56_out_aa(x₁)) = 1 + x₁
POL(merge40_in_aaa) = 1
POL(merge40_out_aaa) = 0
POL(mergesort1_in_ga(x₁)) = x₁
POL(mergesort1_out_ga) = 7
POL(s(x₁)) = x₁
POL(split16_in_gaa(x₁)) = 1 + x₁
POL(split16_out_gaa(x₁, x₂)) = x₁ + x₂

(47) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

split16_in_gaa(.(T57, .(T58, T59))) → U1_gaa(T57, T58, split16_in_gaa(T59))
U1_gaa(T57, T58, split16_out_gaa(X167, X168)) → split16_out_gaa(.(T57, X167), .(T58, X168))
merge40_in_aaa → U2_aaa(le56_in_aa)
merge40_in_aaa → U4_aaa(gt73_in_aa)
U2_aaa(le56_out_aa(T103)) → U3_aaa(merge40_in_aaa)
U4_aaa(gt73_out_aa(T146)) → U5_aaa(merge40_in_aaa)
le56_in_aa → U6_aa(le56_in_aa)
le56_in_aa → le56_out_aa(0)
U3_aaa(merge40_out_aaa) → merge40_out_aaa
gt73_in_aa → U7_aa(gt73_in_aa)
gt73_in_aa → gt73_out_aa(0)
U5_aaa(merge40_out_aaa) → merge40_out_aaa
U6_aa(le56_out_aa(T121)) → le56_out_aa(s(T121))
U7_aa(gt73_out_aa(T163)) → gt73_out_aa(s(T163))

The set Q consists of the following terms:

split16_in_gaa(x0)
mergesort1_in_ga(x0)
U1_gaa(x0, x1, x2)
U8_ga(x0)
U9_ga(x0, x1, x2)
U10_ga(x0)
U11_ga(x0, x1, x2)
U12_ga(x0)
U13_ga(x0)
U14_ga(x0)
merge40_in_aaa
U2_aaa(x0)
U4_aaa(x0)
le56_in_aa
U3_aaa(x0)
gt73_in_aa
U5_aaa(x0)
U6_aa(x0)
U7_aa(x0)

We have to consider all (P,Q,R)-chains.

(48) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(49) YES

(50) PrologToPiTRSProof (SOUND transformation)

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U8_ga(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
split16_in_gaa([], [], []) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, []), .(T41, []), []) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_gaa(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
U1_gaa(T57, T58, T59, X167, X168, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, T14, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U9_ga(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U10_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T22, T26), X23)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U12_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T23, T27), X24)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_ga(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_ga(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_ga(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
merge40_in_aaa([], T77, T77) → merge40_out_aaa([], T77, T77)
merge40_in_aaa(T82, [], T82) → merge40_out_aaa(T82, [], T82)
merge40_in_aaa(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_aaa(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_aaa(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
merge40_in_aaa(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_aaa(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_aaa(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U5_aaa(T144, T145, T146, T147, T149, merge40_out_aaa(.(T144, T145), T147, T149)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U3_aaa(T103, T104, T105, T106, T108, merge40_out_aaa(T104, .(T105, T106), T108)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U14_ga(T22, T23, T24, T14, merge40_out_aaa(T63, T68, T14)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(51) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U8_ga(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
split16_in_gaa([], [], []) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, []), .(T41, []), []) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_gaa(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
U1_gaa(T57, T58, T59, X167, X168, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, T14, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U9_ga(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U10_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T22, T26), X23)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U12_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T23, T27), X24)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_ga(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_ga(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_ga(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
merge40_in_aaa([], T77, T77) → merge40_out_aaa([], T77, T77)
merge40_in_aaa(T82, [], T82) → merge40_out_aaa(T82, [], T82)
merge40_in_aaa(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_aaa(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_aaa(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
merge40_in_aaa(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_aaa(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_aaa(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U5_aaa(T144, T145, T146, T147, T149, merge40_out_aaa(.(T144, T145), T147, T149)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U3_aaa(T103, T104, T105, T106, T108, merge40_out_aaa(T104, .(T105, T106), T108)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U14_ga(T22, T23, T24, T14, merge40_out_aaa(T63, T68, T14)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)

(52) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

MERGESORT1_IN_GA(.(T22, .(T23, T24)), T14) → U8_GA(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
MERGESORT1_IN_GA(.(T22, .(T23, T24)), T14) → SPLIT16_IN_GAA(T24, X53, X54)
SPLIT16_IN_GAA(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_GAA(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
SPLIT16_IN_GAA(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → SPLIT16_IN_GAA(T59, X167, X168)
MERGESORT1_IN_GA(.(T22, .(T23, T24)), T14) → U9_GA(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_GA(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_GA(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U9_GA(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → MERGESORT1_IN_GA(.(T22, T26), X23)
U9_GA(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_GA(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_GA(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_GA(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U11_GA(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → MERGESORT1_IN_GA(.(T23, T27), X24)
U11_GA(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_GA(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_GA(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_GA(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
U13_GA(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → MERGE40_IN_AAA(T63, T68, T14)
MERGE40_IN_AAA(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_AAA(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
MERGE40_IN_AAA(.(T103, T104), .(T105, T106), .(T103, T108)) → LE56_IN_AA(T103, T105)
LE56_IN_AA(s(T121), s(T122)) → U6_AA(T121, T122, le56_in_aa(T121, T122))
LE56_IN_AA(s(T121), s(T122)) → LE56_IN_AA(T121, T122)
U2_AAA(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_AAA(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
U2_AAA(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → MERGE40_IN_AAA(T104, .(T105, T106), T108)
MERGE40_IN_AAA(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_AAA(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
MERGE40_IN_AAA(.(T144, T145), .(T146, T147), .(T146, T149)) → GT73_IN_AA(T144, T146)
GT73_IN_AA(s(T162), s(T163)) → U7_AA(T162, T163, gt73_in_aa(T162, T163))
GT73_IN_AA(s(T162), s(T163)) → GT73_IN_AA(T162, T163)
U4_AAA(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_AAA(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U4_AAA(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → MERGE40_IN_AAA(.(T144, T145), T147, T149)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U8_ga(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
split16_in_gaa([], [], []) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, []), .(T41, []), []) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_gaa(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
U1_gaa(T57, T58, T59, X167, X168, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, T14, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U9_ga(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U10_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T22, T26), X23)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U12_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T23, T27), X24)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_ga(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_ga(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_ga(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
merge40_in_aaa([], T77, T77) → merge40_out_aaa([], T77, T77)
merge40_in_aaa(T82, [], T82) → merge40_out_aaa(T82, [], T82)
merge40_in_aaa(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_aaa(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_aaa(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
merge40_in_aaa(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_aaa(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_aaa(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U5_aaa(T144, T145, T146, T147, T149, merge40_out_aaa(.(T144, T145), T147, T149)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U3_aaa(T103, T104, T105, T106, T108, merge40_out_aaa(T104, .(T105, T106), T108)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U14_ga(T22, T23, T24, T14, merge40_out_aaa(T63, T68, T14)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2) = mergesort1_in_ga(x1)
[] = []
mergesort1_out_ga(x1, x2) = mergesort1_out_ga(x1)
.(x1, x2) = .(x1, x2)
U8_ga(x1, x2, x3, x4, x5) = U8_ga(x1, x2, x3, x5)
split16_in_gaa(x1, x2, x3) = split16_in_gaa(x1)
split16_out_gaa(x1, x2, x3) = split16_out_gaa(x1, x2, x3)
U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x2, x3, x6)
U9_ga(x1, x2, x3, x4, x5) = U9_ga(x1, x2, x3, x5)
U10_ga(x1, x2, x3, x4, x5) = U10_ga(x1, x2, x3, x5)
U11_ga(x1, x2, x3, x4, x5, x6) = U11_ga(x1, x2, x3, x5, x6)
U12_ga(x1, x2, x3, x4, x5) = U12_ga(x1, x2, x3, x5)
U13_ga(x1, x2, x3, x4, x5, x6) = U13_ga(x1, x2, x3, x6)
U14_ga(x1, x2, x3, x4, x5) = U14_ga(x1, x2, x3, x5)
merge40_in_aaa(x1, x2, x3) = merge40_in_aaa
merge40_out_aaa(x1, x2, x3) = merge40_out_aaa
U2_aaa(x1, x2, x3, x4, x5, x6) = U2_aaa(x6)
le56_in_aa(x1, x2) = le56_in_aa
U6_aa(x1, x2, x3) = U6_aa(x3)
le56_out_aa(x1, x2) = le56_out_aa(x1)
U3_aaa(x1, x2, x3, x4, x5, x6) = U3_aaa(x6)
U4_aaa(x1, x2, x3, x4, x5, x6) = U4_aaa(x6)
gt73_in_aa(x1, x2) = gt73_in_aa
U7_aa(x1, x2, x3) = U7_aa(x3)
gt73_out_aa(x1, x2) = gt73_out_aa(x2)
U5_aaa(x1, x2, x3, x4, x5, x6) = U5_aaa(x6)
s(x1) = s(x1)
MERGESORT1_IN_GA(x1, x2) = MERGESORT1_IN_GA(x1)
U8_GA(x1, x2, x3, x4, x5) = U8_GA(x1, x2, x3, x5)
SPLIT16_IN_GAA(x1, x2, x3) = SPLIT16_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x1, x2, x3, x6)
U9_GA(x1, x2, x3, x4, x5) = U9_GA(x1, x2, x3, x5)
U10_GA(x1, x2, x3, x4, x5) = U10_GA(x1, x2, x3, x5)
U11_GA(x1, x2, x3, x4, x5, x6) = U11_GA(x1, x2, x3, x5, x6)
U12_GA(x1, x2, x3, x4, x5) = U12_GA(x1, x2, x3, x5)
U13_GA(x1, x2, x3, x4, x5, x6) = U13_GA(x1, x2, x3, x6)
U14_GA(x1, x2, x3, x4, x5) = U14_GA(x1, x2, x3, x5)
MERGE40_IN_AAA(x1, x2, x3) = MERGE40_IN_AAA
U2_AAA(x1, x2, x3, x4, x5, x6) = U2_AAA(x6)
LE56_IN_AA(x1, x2) = LE56_IN_AA
U6_AA(x1, x2, x3) = U6_AA(x3)
U3_AAA(x1, x2, x3, x4, x5, x6) = U3_AAA(x6)
U4_AAA(x1, x2, x3, x4, x5, x6) = U4_AAA(x6)
GT73_IN_AA(x1, x2) = GT73_IN_AA
U7_AA(x1, x2, x3) = U7_AA(x3)
U5_AAA(x1, x2, x3, x4, x5, x6) = U5_AAA(x6)

We have to consider all (P,R,Pi)-chains

(53) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MERGESORT1_IN_GA(.(T22, .(T23, T24)), T14) → U8_GA(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
MERGESORT1_IN_GA(.(T22, .(T23, T24)), T14) → SPLIT16_IN_GAA(T24, X53, X54)
SPLIT16_IN_GAA(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_GAA(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
SPLIT16_IN_GAA(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → SPLIT16_IN_GAA(T59, X167, X168)
MERGESORT1_IN_GA(.(T22, .(T23, T24)), T14) → U9_GA(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_GA(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_GA(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U9_GA(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → MERGESORT1_IN_GA(.(T22, T26), X23)
U9_GA(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_GA(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_GA(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_GA(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U11_GA(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → MERGESORT1_IN_GA(.(T23, T27), X24)
U11_GA(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_GA(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_GA(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_GA(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
U13_GA(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → MERGE40_IN_AAA(T63, T68, T14)
MERGE40_IN_AAA(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_AAA(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
MERGE40_IN_AAA(.(T103, T104), .(T105, T106), .(T103, T108)) → LE56_IN_AA(T103, T105)
LE56_IN_AA(s(T121), s(T122)) → U6_AA(T121, T122, le56_in_aa(T121, T122))
LE56_IN_AA(s(T121), s(T122)) → LE56_IN_AA(T121, T122)
U2_AAA(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_AAA(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
U2_AAA(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → MERGE40_IN_AAA(T104, .(T105, T106), T108)
MERGE40_IN_AAA(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_AAA(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
MERGE40_IN_AAA(.(T144, T145), .(T146, T147), .(T146, T149)) → GT73_IN_AA(T144, T146)
GT73_IN_AA(s(T162), s(T163)) → U7_AA(T162, T163, gt73_in_aa(T162, T163))
GT73_IN_AA(s(T162), s(T163)) → GT73_IN_AA(T162, T163)
U4_AAA(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_AAA(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U4_AAA(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → MERGE40_IN_AAA(.(T144, T145), T147, T149)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U8_ga(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
split16_in_gaa([], [], []) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, []), .(T41, []), []) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_gaa(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
U1_gaa(T57, T58, T59, X167, X168, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, T14, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U9_ga(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U10_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T22, T26), X23)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U12_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T23, T27), X24)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_ga(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_ga(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_ga(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
merge40_in_aaa([], T77, T77) → merge40_out_aaa([], T77, T77)
merge40_in_aaa(T82, [], T82) → merge40_out_aaa(T82, [], T82)
merge40_in_aaa(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_aaa(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_aaa(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
merge40_in_aaa(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_aaa(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_aaa(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U5_aaa(T144, T145, T146, T147, T149, merge40_out_aaa(.(T144, T145), T147, T149)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U3_aaa(T103, T104, T105, T106, T108, merge40_out_aaa(T104, .(T105, T106), T108)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U14_ga(T22, T23, T24, T14, merge40_out_aaa(T63, T68, T14)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)

(54) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 14 less nodes.

(55) Complex Obligation (AND)

(56) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

GT73_IN_AA(s(T162), s(T163)) → GT73_IN_AA(T162, T163)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U8_ga(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
split16_in_gaa([], [], []) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, []), .(T41, []), []) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_gaa(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
U1_gaa(T57, T58, T59, X167, X168, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, T14, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U9_ga(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U10_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T22, T26), X23)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U12_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T23, T27), X24)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_ga(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_ga(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_ga(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
merge40_in_aaa([], T77, T77) → merge40_out_aaa([], T77, T77)
merge40_in_aaa(T82, [], T82) → merge40_out_aaa(T82, [], T82)
merge40_in_aaa(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_aaa(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_aaa(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
merge40_in_aaa(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_aaa(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_aaa(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U5_aaa(T144, T145, T146, T147, T149, merge40_out_aaa(.(T144, T145), T147, T149)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U3_aaa(T103, T104, T105, T106, T108, merge40_out_aaa(T104, .(T105, T106), T108)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U14_ga(T22, T23, T24, T14, merge40_out_aaa(T63, T68, T14)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)

(57) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(58) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

GT73_IN_AA(s(T162), s(T163)) → GT73_IN_AA(T162, T163)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
GT73_IN_AA(x1, x2) = GT73_IN_AA

We have to consider all (P,R,Pi)-chains

(59) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(60) Obligation:

Q DP problem:
The TRS P consists of the following rules:

GT73_IN_AA → GT73_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(61) NonTerminationProof (EQUIVALENT transformation)

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from GT73_IN_AA to GT73_IN_AA.

(62) NO

(63) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LE56_IN_AA(s(T121), s(T122)) → LE56_IN_AA(T121, T122)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U8_ga(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
split16_in_gaa([], [], []) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, []), .(T41, []), []) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_gaa(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
U1_gaa(T57, T58, T59, X167, X168, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, T14, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U9_ga(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U10_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T22, T26), X23)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U12_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T23, T27), X24)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_ga(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_ga(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_ga(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
merge40_in_aaa([], T77, T77) → merge40_out_aaa([], T77, T77)
merge40_in_aaa(T82, [], T82) → merge40_out_aaa(T82, [], T82)
merge40_in_aaa(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_aaa(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_aaa(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
merge40_in_aaa(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_aaa(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_aaa(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U5_aaa(T144, T145, T146, T147, T149, merge40_out_aaa(.(T144, T145), T147, T149)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U3_aaa(T103, T104, T105, T106, T108, merge40_out_aaa(T104, .(T105, T106), T108)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U14_ga(T22, T23, T24, T14, merge40_out_aaa(T63, T68, T14)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)

(64) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(65) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LE56_IN_AA(s(T121), s(T122)) → LE56_IN_AA(T121, T122)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
LE56_IN_AA(x1, x2) = LE56_IN_AA

We have to consider all (P,R,Pi)-chains

(66) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(67) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LE56_IN_AA → LE56_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(68) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LE56_IN_AA to LE56_IN_AA.

(69) NO

(70) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

U2_AAA(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → MERGE40_IN_AAA(T104, .(T105, T106), T108)
MERGE40_IN_AAA(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_AAA(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
MERGE40_IN_AAA(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_AAA(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
U4_AAA(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → MERGE40_IN_AAA(.(T144, T145), T147, T149)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U8_ga(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
split16_in_gaa([], [], []) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, []), .(T41, []), []) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_gaa(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
U1_gaa(T57, T58, T59, X167, X168, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, T14, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U9_ga(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U10_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T22, T26), X23)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U12_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T23, T27), X24)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_ga(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_ga(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_ga(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
merge40_in_aaa([], T77, T77) → merge40_out_aaa([], T77, T77)
merge40_in_aaa(T82, [], T82) → merge40_out_aaa(T82, [], T82)
merge40_in_aaa(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_aaa(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_aaa(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
merge40_in_aaa(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_aaa(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_aaa(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U5_aaa(T144, T145, T146, T147, T149, merge40_out_aaa(.(T144, T145), T147, T149)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U3_aaa(T103, T104, T105, T106, T108, merge40_out_aaa(T104, .(T105, T106), T108)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U14_ga(T22, T23, T24, T14, merge40_out_aaa(T63, T68, T14)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)

(71) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(72) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

U2_AAA(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → MERGE40_IN_AAA(T104, .(T105, T106), T108)
MERGE40_IN_AAA(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_AAA(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
MERGE40_IN_AAA(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_AAA(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
U4_AAA(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → MERGE40_IN_AAA(.(T144, T145), T147, T149)

The TRS R consists of the following rules:

le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))

(73) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(74) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U2_AAA(le56_out_aa(T103)) → MERGE40_IN_AAA
MERGE40_IN_AAA → U2_AAA(le56_in_aa)
MERGE40_IN_AAA → U4_AAA(gt73_in_aa)
U4_AAA(gt73_out_aa(T146)) → MERGE40_IN_AAA

The TRS R consists of the following rules:

le56_in_aa → U6_aa(le56_in_aa)
le56_in_aa → le56_out_aa(0)
gt73_in_aa → U7_aa(gt73_in_aa)
gt73_in_aa → gt73_out_aa(0)
U6_aa(le56_out_aa(T121)) → le56_out_aa(s(T121))
U7_aa(gt73_out_aa(T163)) → gt73_out_aa(s(T163))

The set Q consists of the following terms:

le56_in_aa
gt73_in_aa
U6_aa(x0)
U7_aa(x0)

We have to consider all (P,Q,R)-chains.

(75) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MERGE40_IN_AAA → U2_AAA(le56_in_aa) at position [0] we obtained the following new rules [LPAR04]:

MERGE40_IN_AAA → U2_AAA(U6_aa(le56_in_aa))
MERGE40_IN_AAA → U2_AAA(le56_out_aa(0))

(76) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U2_AAA(le56_out_aa(T103)) → MERGE40_IN_AAA
MERGE40_IN_AAA → U4_AAA(gt73_in_aa)
U4_AAA(gt73_out_aa(T146)) → MERGE40_IN_AAA
MERGE40_IN_AAA → U2_AAA(U6_aa(le56_in_aa))
MERGE40_IN_AAA → U2_AAA(le56_out_aa(0))

The TRS R consists of the following rules:

le56_in_aa → U6_aa(le56_in_aa)
le56_in_aa → le56_out_aa(0)
gt73_in_aa → U7_aa(gt73_in_aa)
gt73_in_aa → gt73_out_aa(0)
U6_aa(le56_out_aa(T121)) → le56_out_aa(s(T121))
U7_aa(gt73_out_aa(T163)) → gt73_out_aa(s(T163))

The set Q consists of the following terms:

le56_in_aa
gt73_in_aa
U6_aa(x0)
U7_aa(x0)

We have to consider all (P,Q,R)-chains.

(77) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MERGE40_IN_AAA → U4_AAA(gt73_in_aa) at position [0] we obtained the following new rules [LPAR04]:

MERGE40_IN_AAA → U4_AAA(U7_aa(gt73_in_aa))
MERGE40_IN_AAA → U4_AAA(gt73_out_aa(0))

(78) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U2_AAA(le56_out_aa(T103)) → MERGE40_IN_AAA
U4_AAA(gt73_out_aa(T146)) → MERGE40_IN_AAA
MERGE40_IN_AAA → U2_AAA(U6_aa(le56_in_aa))
MERGE40_IN_AAA → U2_AAA(le56_out_aa(0))
MERGE40_IN_AAA → U4_AAA(U7_aa(gt73_in_aa))
MERGE40_IN_AAA → U4_AAA(gt73_out_aa(0))

The TRS R consists of the following rules:

le56_in_aa → U6_aa(le56_in_aa)
le56_in_aa → le56_out_aa(0)
gt73_in_aa → U7_aa(gt73_in_aa)
gt73_in_aa → gt73_out_aa(0)
U6_aa(le56_out_aa(T121)) → le56_out_aa(s(T121))
U7_aa(gt73_out_aa(T163)) → gt73_out_aa(s(T163))

The set Q consists of the following terms:

le56_in_aa
gt73_in_aa
U6_aa(x0)
U7_aa(x0)

We have to consider all (P,Q,R)-chains.

(79) NonTerminationProof (EQUIVALENT transformation)

Matcher: [ ]
Semiunifier: [ ]

(80) NO

(81) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SPLIT16_IN_GAA(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → SPLIT16_IN_GAA(T59, X167, X168)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U8_ga(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
split16_in_gaa([], [], []) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, []), .(T41, []), []) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_gaa(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
U1_gaa(T57, T58, T59, X167, X168, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, T14, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U9_ga(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U10_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T22, T26), X23)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U12_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T23, T27), X24)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_ga(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_ga(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_ga(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
merge40_in_aaa([], T77, T77) → merge40_out_aaa([], T77, T77)
merge40_in_aaa(T82, [], T82) → merge40_out_aaa(T82, [], T82)
merge40_in_aaa(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_aaa(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_aaa(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
merge40_in_aaa(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_aaa(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_aaa(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U5_aaa(T144, T145, T146, T147, T149, merge40_out_aaa(.(T144, T145), T147, T149)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U3_aaa(T103, T104, T105, T106, T108, merge40_out_aaa(T104, .(T105, T106), T108)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U14_ga(T22, T23, T24, T14, merge40_out_aaa(T63, T68, T14)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)

(82) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(83) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SPLIT16_IN_GAA(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → SPLIT16_IN_GAA(T59, X167, X168)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
SPLIT16_IN_GAA(x1, x2, x3) = SPLIT16_IN_GAA(x1)

We have to consider all (P,R,Pi)-chains

(84) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(85) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SPLIT16_IN_GAA(.(T57, .(T58, T59))) → SPLIT16_IN_GAA(T59)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(86) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

SPLIT16_IN_GAA(.(T57, .(T58, T59))) → SPLIT16_IN_GAA(T59)
The graph contains the following edges 1 > 1

(87) YES

(88) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MERGESORT1_IN_GA(.(T22, .(T23, T24)), T14) → U9_GA(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_GA(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → MERGESORT1_IN_GA(.(T22, T26), X23)
U9_GA(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_GA(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_GA(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → MERGESORT1_IN_GA(.(T23, T27), X24)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U8_ga(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
split16_in_gaa([], [], []) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, []), .(T41, []), []) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_gaa(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
U1_gaa(T57, T58, T59, X167, X168, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, T14, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U9_ga(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U10_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T22, T26), X23)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U12_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T23, T27), X24)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_ga(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U13_ga(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_ga(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
merge40_in_aaa([], T77, T77) → merge40_out_aaa([], T77, T77)
merge40_in_aaa(T82, [], T82) → merge40_out_aaa(T82, [], T82)
merge40_in_aaa(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_aaa(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_aaa(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
merge40_in_aaa(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_aaa(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_aaa(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
U5_aaa(T144, T145, T146, T147, T149, merge40_out_aaa(.(T144, T145), T147, T149)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U3_aaa(T103, T104, T105, T106, T108, merge40_out_aaa(T104, .(T105, T106), T108)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U14_ga(T22, T23, T24, T14, merge40_out_aaa(T63, T68, T14)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)

(89) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(90) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MERGESORT1_IN_GA(.(T22, .(T23, T24)), T14) → U9_GA(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U9_GA(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → MERGESORT1_IN_GA(.(T22, T26), X23)
U9_GA(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_GA(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U11_GA(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → MERGESORT1_IN_GA(.(T23, T27), X24)

The TRS R consists of the following rules:

split16_in_gaa([], [], []) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, []), .(T41, []), []) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168)) → U1_gaa(T57, T58, T59, X167, X168, split16_in_gaa(T59, X167, X168))
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U8_ga(T22, T23, T24, T14, split16_in_gaa(T24, X53, X54))
mergesort1_in_ga(.(T22, .(T23, T24)), T14) → U9_ga(T22, T23, T24, T14, split16_in_gaa(T24, T26, T27))
U1_gaa(T57, T58, T59, X167, X168, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, T14, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T22, T26), X23))
U9_ga(T22, T23, T24, T14, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T14, T27, mergesort1_in_ga(.(T22, T26), T63))
U10_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T22, T26), X23)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U12_ga(T22, T23, T24, T14, mergesort1_in_ga(.(T23, T27), X24))
U11_ga(T22, T23, T24, T14, T27, mergesort1_out_ga(.(T22, T26), T63)) → U13_ga(T22, T23, T24, T14, T63, mergesort1_in_ga(.(T23, T27), T68))
U12_ga(T22, T23, T24, T14, mergesort1_out_ga(.(T23, T27), X24)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
U13_ga(T22, T23, T24, T14, T63, mergesort1_out_ga(.(T23, T27), T68)) → U14_ga(T22, T23, T24, T14, merge40_in_aaa(T63, T68, T14))
U14_ga(T22, T23, T24, T14, merge40_out_aaa(T63, T68, T14)) → mergesort1_out_ga(.(T22, .(T23, T24)), T14)
merge40_in_aaa([], T77, T77) → merge40_out_aaa([], T77, T77)
merge40_in_aaa(T82, [], T82) → merge40_out_aaa(T82, [], T82)
merge40_in_aaa(.(T103, T104), .(T105, T106), .(T103, T108)) → U2_aaa(T103, T104, T105, T106, T108, le56_in_aa(T103, T105))
merge40_in_aaa(.(T144, T145), .(T146, T147), .(T146, T149)) → U4_aaa(T144, T145, T146, T147, T149, gt73_in_aa(T144, T146))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
U2_aaa(T103, T104, T105, T106, T108, le56_out_aa(T103, T105)) → U3_aaa(T103, T104, T105, T106, T108, merge40_in_aaa(T104, .(T105, T106), T108))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U4_aaa(T144, T145, T146, T147, T149, gt73_out_aa(T144, T146)) → U5_aaa(T144, T145, T146, T147, T149, merge40_in_aaa(.(T144, T145), T147, T149))
le56_in_aa(s(T121), s(T122)) → U6_aa(T121, T122, le56_in_aa(T121, T122))
le56_in_aa(0, s(T129)) → le56_out_aa(0, s(T129))
le56_in_aa(0, 0) → le56_out_aa(0, 0)
U3_aaa(T103, T104, T105, T106, T108, merge40_out_aaa(T104, .(T105, T106), T108)) → merge40_out_aaa(.(T103, T104), .(T105, T106), .(T103, T108))
gt73_in_aa(s(T162), s(T163)) → U7_aa(T162, T163, gt73_in_aa(T162, T163))
gt73_in_aa(s(T168), 0) → gt73_out_aa(s(T168), 0)
U5_aaa(T144, T145, T146, T147, T149, merge40_out_aaa(.(T144, T145), T147, T149)) → merge40_out_aaa(.(T144, T145), .(T146, T147), .(T146, T149))
U6_aa(T121, T122, le56_out_aa(T121, T122)) → le56_out_aa(s(T121), s(T122))
U7_aa(T162, T163, gt73_out_aa(T162, T163)) → gt73_out_aa(s(T162), s(T163))

(91) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(92) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MERGESORT1_IN_GA(.(T22, .(T23, T24))) → U9_GA(T22, T23, T24, split16_in_gaa(T24))
U9_GA(T22, T23, T24, split16_out_gaa(T24, T26, T27)) → MERGESORT1_IN_GA(.(T22, T26))
U9_GA(T22, T23, T24, split16_out_gaa(T24, T26, T27)) → U11_GA(T22, T23, T24, T27, mergesort1_in_ga(.(T22, T26)))
U11_GA(T22, T23, T24, T27, mergesort1_out_ga(.(T22, T26))) → MERGESORT1_IN_GA(.(T23, T27))

The TRS R consists of the following rules:

split16_in_gaa([]) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, [])) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59))) → U1_gaa(T57, T58, T59, split16_in_gaa(T59))
mergesort1_in_ga(.(T4, [])) → mergesort1_out_ga(.(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24))) → U8_ga(T22, T23, T24, split16_in_gaa(T24))
mergesort1_in_ga(.(T22, .(T23, T24))) → U9_ga(T22, T23, T24, split16_in_gaa(T24))
U1_gaa(T57, T58, T59, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)))
U9_ga(T22, T23, T24, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, mergesort1_in_ga(.(T22, T26)))
U9_ga(T22, T23, T24, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T27, mergesort1_in_ga(.(T22, T26)))
U10_ga(T22, T23, T24, mergesort1_out_ga(.(T22, T26))) → mergesort1_out_ga(.(T22, .(T23, T24)))
U11_ga(T22, T23, T24, T27, mergesort1_out_ga(.(T22, T26))) → U12_ga(T22, T23, T24, mergesort1_in_ga(.(T23, T27)))
U11_ga(T22, T23, T24, T27, mergesort1_out_ga(.(T22, T26))) → U13_ga(T22, T23, T24, mergesort1_in_ga(.(T23, T27)))
U12_ga(T22, T23, T24, mergesort1_out_ga(.(T23, T27))) → mergesort1_out_ga(.(T22, .(T23, T24)))
U13_ga(T22, T23, T24, mergesort1_out_ga(.(T23, T27))) → U14_ga(T22, T23, T24, merge40_in_aaa)
U14_ga(T22, T23, T24, merge40_out_aaa) → mergesort1_out_ga(.(T22, .(T23, T24)))
merge40_in_aaa → merge40_out_aaa
merge40_in_aaa → U2_aaa(le56_in_aa)
merge40_in_aaa → U4_aaa(gt73_in_aa)
U2_aaa(le56_out_aa(T103)) → merge40_out_aaa
U2_aaa(le56_out_aa(T103)) → U3_aaa(merge40_in_aaa)
U4_aaa(gt73_out_aa(T146)) → merge40_out_aaa
U4_aaa(gt73_out_aa(T146)) → U5_aaa(merge40_in_aaa)
le56_in_aa → U6_aa(le56_in_aa)
le56_in_aa → le56_out_aa(0)
U3_aaa(merge40_out_aaa) → merge40_out_aaa
gt73_in_aa → U7_aa(gt73_in_aa)
gt73_in_aa → gt73_out_aa(0)
U5_aaa(merge40_out_aaa) → merge40_out_aaa
U6_aa(le56_out_aa(T121)) → le56_out_aa(s(T121))
U7_aa(gt73_out_aa(T163)) → gt73_out_aa(s(T163))

The set Q consists of the following terms:

split16_in_gaa(x0)
mergesort1_in_ga(x0)
U1_gaa(x0, x1, x2, x3)
U8_ga(x0, x1, x2, x3)
U9_ga(x0, x1, x2, x3)
U10_ga(x0, x1, x2, x3)
U11_ga(x0, x1, x2, x3, x4)
U12_ga(x0, x1, x2, x3)
U13_ga(x0, x1, x2, x3)
U14_ga(x0, x1, x2, x3)
merge40_in_aaa
U2_aaa(x0)
U4_aaa(x0)
le56_in_aa
U3_aaa(x0)
gt73_in_aa
U5_aaa(x0)
U6_aa(x0)
U7_aa(x0)

We have to consider all (P,Q,R)-chains.

(93) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11_GA(T22, T23, T24, T27, mergesort1_out_ga(.(T22, T26))) → MERGESORT1_IN_GA(.(T23, T27))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(MERGESORT1_IN_GA(x₁)) = 0 +
[ 1, 0 ]

· x₁

POL(.(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 0 1 /

· x₂

POL(U9_GA(x₁, x₂, x₃, x₄)) = 0 +
[ 0, 0 ]

· x₁ +
[ 0, 0 ]

· x₂ +
[ 0, 0 ]

· x₃ +
[ 0, 1 ]

· x₄

POL(split16_in_gaa(x₁)) =
/ 1 \

\ 1 /

+
/ 0 0 \

\ 0 1 /

· x₁

POL(split16_out_gaa(x₁, x₂, x₃)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 1 /

· x₂ +
/ 0 0 \

\ 0 1 /

· x₃

POL(U11_GA(x₁, x₂, x₃, x₄, x₅)) = 0 +
[ 0, 0 ]

· x₁ +
[ 0, 0 ]

· x₂ +
[ 0, 0 ]

· x₃ +
[ 0, 1 ]

· x₄ +
[ 1, 0 ]

· x₅

POL(mergesort1_in_ga(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁

POL(mergesort1_out_ga(x₁)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL([]) =
/ 0 \

\ 1 /

POL(U1_gaa(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 1 1 /

· x₄

POL(U8_ga(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 1 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U9_ga(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 1 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U10_ga(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 1 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U11_ga(x₁, x₂, x₃, x₄, x₅)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 1 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄ +
/ 0 0 \

\ 0 0 /

· x₅

POL(U12_ga(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U13_ga(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 1 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U14_ga(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(merge40_in_aaa) =
/ 0 \

\ 0 /

POL(merge40_out_aaa) =
/ 1 \

\ 1 /

POL(U2_aaa(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(le56_in_aa) =
/ 0 \

\ 1 /

POL(U4_aaa(x₁)) =
/ 1 \

\ 0 /

+
/ 1 1 \

\ 0 0 /

· x₁

POL(gt73_in_aa) =
/ 0 \

\ 1 /

POL(U6_aa(x₁)) =
/ 1 \

\ 0 /

+
/ 1 1 \

\ 0 0 /

· x₁

POL(le56_out_aa(x₁)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 1 /

· x₁

POL(0) =
/ 0 \

\ 0 /

POL(U3_aaa(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(U7_aa(x₁)) =
/ 0 \

\ 1 /

+
/ 1 1 \

\ 1 1 /

· x₁

POL(gt73_out_aa(x₁)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 1 0 /

· x₁

POL(U5_aaa(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 1 1 /

· x₁

POL(s(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 1 0 /

· x₁

The following usable rules [FROCOS05] were oriented:

split16_in_gaa([]) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, [])) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59))) → U1_gaa(T57, T58, T59, split16_in_gaa(T59))
mergesort1_in_ga(.(T4, [])) → mergesort1_out_ga(.(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24))) → U8_ga(T22, T23, T24, split16_in_gaa(T24))
mergesort1_in_ga(.(T22, .(T23, T24))) → U9_ga(T22, T23, T24, split16_in_gaa(T24))
U9_ga(T22, T23, T24, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, mergesort1_in_ga(.(T22, T26)))
U10_ga(T22, T23, T24, mergesort1_out_ga(.(T22, T26))) → mergesort1_out_ga(.(T22, .(T23, T24)))
U9_ga(T22, T23, T24, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T27, mergesort1_in_ga(.(T22, T26)))
U11_ga(T22, T23, T24, T27, mergesort1_out_ga(.(T22, T26))) → U12_ga(T22, T23, T24, mergesort1_in_ga(.(T23, T27)))
U12_ga(T22, T23, T24, mergesort1_out_ga(.(T23, T27))) → mergesort1_out_ga(.(T22, .(T23, T24)))
U11_ga(T22, T23, T24, T27, mergesort1_out_ga(.(T22, T26))) → U13_ga(T22, T23, T24, mergesort1_in_ga(.(T23, T27)))
U13_ga(T22, T23, T24, mergesort1_out_ga(.(T23, T27))) → U14_ga(T22, T23, T24, merge40_in_aaa)
U8_ga(T22, T23, T24, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)))
U1_gaa(T57, T58, T59, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U14_ga(T22, T23, T24, merge40_out_aaa) → mergesort1_out_ga(.(T22, .(T23, T24)))

(94) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MERGESORT1_IN_GA(.(T22, .(T23, T24))) → U9_GA(T22, T23, T24, split16_in_gaa(T24))
U9_GA(T22, T23, T24, split16_out_gaa(T24, T26, T27)) → MERGESORT1_IN_GA(.(T22, T26))
U9_GA(T22, T23, T24, split16_out_gaa(T24, T26, T27)) → U11_GA(T22, T23, T24, T27, mergesort1_in_ga(.(T22, T26)))

The TRS R consists of the following rules:

split16_in_gaa([]) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, [])) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59))) → U1_gaa(T57, T58, T59, split16_in_gaa(T59))
mergesort1_in_ga(.(T4, [])) → mergesort1_out_ga(.(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24))) → U8_ga(T22, T23, T24, split16_in_gaa(T24))
mergesort1_in_ga(.(T22, .(T23, T24))) → U9_ga(T22, T23, T24, split16_in_gaa(T24))
U1_gaa(T57, T58, T59, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)))
U9_ga(T22, T23, T24, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, mergesort1_in_ga(.(T22, T26)))
U9_ga(T22, T23, T24, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T27, mergesort1_in_ga(.(T22, T26)))
U10_ga(T22, T23, T24, mergesort1_out_ga(.(T22, T26))) → mergesort1_out_ga(.(T22, .(T23, T24)))
U11_ga(T22, T23, T24, T27, mergesort1_out_ga(.(T22, T26))) → U12_ga(T22, T23, T24, mergesort1_in_ga(.(T23, T27)))
U11_ga(T22, T23, T24, T27, mergesort1_out_ga(.(T22, T26))) → U13_ga(T22, T23, T24, mergesort1_in_ga(.(T23, T27)))
U12_ga(T22, T23, T24, mergesort1_out_ga(.(T23, T27))) → mergesort1_out_ga(.(T22, .(T23, T24)))
U13_ga(T22, T23, T24, mergesort1_out_ga(.(T23, T27))) → U14_ga(T22, T23, T24, merge40_in_aaa)
U14_ga(T22, T23, T24, merge40_out_aaa) → mergesort1_out_ga(.(T22, .(T23, T24)))
merge40_in_aaa → merge40_out_aaa
merge40_in_aaa → U2_aaa(le56_in_aa)
merge40_in_aaa → U4_aaa(gt73_in_aa)
U2_aaa(le56_out_aa(T103)) → merge40_out_aaa
U2_aaa(le56_out_aa(T103)) → U3_aaa(merge40_in_aaa)
U4_aaa(gt73_out_aa(T146)) → merge40_out_aaa
U4_aaa(gt73_out_aa(T146)) → U5_aaa(merge40_in_aaa)
le56_in_aa → U6_aa(le56_in_aa)
le56_in_aa → le56_out_aa(0)
U3_aaa(merge40_out_aaa) → merge40_out_aaa
gt73_in_aa → U7_aa(gt73_in_aa)
gt73_in_aa → gt73_out_aa(0)
U5_aaa(merge40_out_aaa) → merge40_out_aaa
U6_aa(le56_out_aa(T121)) → le56_out_aa(s(T121))
U7_aa(gt73_out_aa(T163)) → gt73_out_aa(s(T163))

The set Q consists of the following terms:

split16_in_gaa(x0)
mergesort1_in_ga(x0)
U1_gaa(x0, x1, x2, x3)
U8_ga(x0, x1, x2, x3)
U9_ga(x0, x1, x2, x3)
U10_ga(x0, x1, x2, x3)
U11_ga(x0, x1, x2, x3, x4)
U12_ga(x0, x1, x2, x3)
U13_ga(x0, x1, x2, x3)
U14_ga(x0, x1, x2, x3)
merge40_in_aaa
U2_aaa(x0)
U4_aaa(x0)
le56_in_aa
U3_aaa(x0)
gt73_in_aa
U5_aaa(x0)
U6_aa(x0)
U7_aa(x0)

We have to consider all (P,Q,R)-chains.

(95) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(96) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U9_GA(T22, T23, T24, split16_out_gaa(T24, T26, T27)) → MERGESORT1_IN_GA(.(T22, T26))
MERGESORT1_IN_GA(.(T22, .(T23, T24))) → U9_GA(T22, T23, T24, split16_in_gaa(T24))

The TRS R consists of the following rules:

split16_in_gaa([]) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, [])) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59))) → U1_gaa(T57, T58, T59, split16_in_gaa(T59))
mergesort1_in_ga(.(T4, [])) → mergesort1_out_ga(.(T4, []))
mergesort1_in_ga(.(T22, .(T23, T24))) → U8_ga(T22, T23, T24, split16_in_gaa(T24))
mergesort1_in_ga(.(T22, .(T23, T24))) → U9_ga(T22, T23, T24, split16_in_gaa(T24))
U1_gaa(T57, T58, T59, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))
U8_ga(T22, T23, T24, split16_out_gaa(T24, X53, X54)) → mergesort1_out_ga(.(T22, .(T23, T24)))
U9_ga(T22, T23, T24, split16_out_gaa(T24, T26, T27)) → U10_ga(T22, T23, T24, mergesort1_in_ga(.(T22, T26)))
U9_ga(T22, T23, T24, split16_out_gaa(T24, T26, T27)) → U11_ga(T22, T23, T24, T27, mergesort1_in_ga(.(T22, T26)))
U10_ga(T22, T23, T24, mergesort1_out_ga(.(T22, T26))) → mergesort1_out_ga(.(T22, .(T23, T24)))
U11_ga(T22, T23, T24, T27, mergesort1_out_ga(.(T22, T26))) → U12_ga(T22, T23, T24, mergesort1_in_ga(.(T23, T27)))
U11_ga(T22, T23, T24, T27, mergesort1_out_ga(.(T22, T26))) → U13_ga(T22, T23, T24, mergesort1_in_ga(.(T23, T27)))
U12_ga(T22, T23, T24, mergesort1_out_ga(.(T23, T27))) → mergesort1_out_ga(.(T22, .(T23, T24)))
U13_ga(T22, T23, T24, mergesort1_out_ga(.(T23, T27))) → U14_ga(T22, T23, T24, merge40_in_aaa)
U14_ga(T22, T23, T24, merge40_out_aaa) → mergesort1_out_ga(.(T22, .(T23, T24)))
merge40_in_aaa → merge40_out_aaa
merge40_in_aaa → U2_aaa(le56_in_aa)
merge40_in_aaa → U4_aaa(gt73_in_aa)
U2_aaa(le56_out_aa(T103)) → merge40_out_aaa
U2_aaa(le56_out_aa(T103)) → U3_aaa(merge40_in_aaa)
U4_aaa(gt73_out_aa(T146)) → merge40_out_aaa
U4_aaa(gt73_out_aa(T146)) → U5_aaa(merge40_in_aaa)
le56_in_aa → U6_aa(le56_in_aa)
le56_in_aa → le56_out_aa(0)
U3_aaa(merge40_out_aaa) → merge40_out_aaa
gt73_in_aa → U7_aa(gt73_in_aa)
gt73_in_aa → gt73_out_aa(0)
U5_aaa(merge40_out_aaa) → merge40_out_aaa
U6_aa(le56_out_aa(T121)) → le56_out_aa(s(T121))
U7_aa(gt73_out_aa(T163)) → gt73_out_aa(s(T163))

The set Q consists of the following terms:

split16_in_gaa(x0)
mergesort1_in_ga(x0)
U1_gaa(x0, x1, x2, x3)
U8_ga(x0, x1, x2, x3)
U9_ga(x0, x1, x2, x3)
U10_ga(x0, x1, x2, x3)
U11_ga(x0, x1, x2, x3, x4)
U12_ga(x0, x1, x2, x3)
U13_ga(x0, x1, x2, x3)
U14_ga(x0, x1, x2, x3)
merge40_in_aaa
U2_aaa(x0)
U4_aaa(x0)
le56_in_aa
U3_aaa(x0)
gt73_in_aa
U5_aaa(x0)
U6_aa(x0)
U7_aa(x0)

We have to consider all (P,Q,R)-chains.

(97) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(98) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U9_GA(T22, T23, T24, split16_out_gaa(T24, T26, T27)) → MERGESORT1_IN_GA(.(T22, T26))
MERGESORT1_IN_GA(.(T22, .(T23, T24))) → U9_GA(T22, T23, T24, split16_in_gaa(T24))

The TRS R consists of the following rules:

split16_in_gaa([]) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, [])) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59))) → U1_gaa(T57, T58, T59, split16_in_gaa(T59))
U1_gaa(T57, T58, T59, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))

The set Q consists of the following terms:

split16_in_gaa(x0)
mergesort1_in_ga(x0)
U1_gaa(x0, x1, x2, x3)
U8_ga(x0, x1, x2, x3)
U9_ga(x0, x1, x2, x3)
U10_ga(x0, x1, x2, x3)
U11_ga(x0, x1, x2, x3, x4)
U12_ga(x0, x1, x2, x3)
U13_ga(x0, x1, x2, x3)
U14_ga(x0, x1, x2, x3)
merge40_in_aaa
U2_aaa(x0)
U4_aaa(x0)
le56_in_aa
U3_aaa(x0)
gt73_in_aa
U5_aaa(x0)
U6_aa(x0)
U7_aa(x0)

We have to consider all (P,Q,R)-chains.

(99) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

mergesort1_in_ga(x0)
U8_ga(x0, x1, x2, x3)
U9_ga(x0, x1, x2, x3)
U10_ga(x0, x1, x2, x3)
U11_ga(x0, x1, x2, x3, x4)
U12_ga(x0, x1, x2, x3)
U13_ga(x0, x1, x2, x3)
U14_ga(x0, x1, x2, x3)
merge40_in_aaa
U2_aaa(x0)
U4_aaa(x0)
le56_in_aa
U3_aaa(x0)
gt73_in_aa
U5_aaa(x0)
U6_aa(x0)
U7_aa(x0)

(100) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U9_GA(T22, T23, T24, split16_out_gaa(T24, T26, T27)) → MERGESORT1_IN_GA(.(T22, T26))
MERGESORT1_IN_GA(.(T22, .(T23, T24))) → U9_GA(T22, T23, T24, split16_in_gaa(T24))

The TRS R consists of the following rules:

split16_in_gaa([]) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, [])) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59))) → U1_gaa(T57, T58, T59, split16_in_gaa(T59))
U1_gaa(T57, T58, T59, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))

The set Q consists of the following terms:

split16_in_gaa(x0)
U1_gaa(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.

(101) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MERGESORT1_IN_GA(.(T22, .(T23, T24))) → U9_GA(T22, T23, T24, split16_in_gaa(T24))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₁ + x₂
POL(MERGESORT1_IN_GA(x₁)) = x₁
POL(U1_gaa(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₂ + x₄
POL(U9_GA(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₄
POL([]) = 0
POL(split16_in_gaa(x₁)) = x₁
POL(split16_out_gaa(x₁, x₂, x₃)) = x₂

The following usable rules [FROCOS05] were oriented:

split16_in_gaa([]) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, [])) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59))) → U1_gaa(T57, T58, T59, split16_in_gaa(T59))
U1_gaa(T57, T58, T59, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))

(102) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U9_GA(T22, T23, T24, split16_out_gaa(T24, T26, T27)) → MERGESORT1_IN_GA(.(T22, T26))

The TRS R consists of the following rules:

split16_in_gaa([]) → split16_out_gaa([], [], [])
split16_in_gaa(.(T41, [])) → split16_out_gaa(.(T41, []), .(T41, []), [])
split16_in_gaa(.(T57, .(T58, T59))) → U1_gaa(T57, T58, T59, split16_in_gaa(T59))
U1_gaa(T57, T58, T59, split16_out_gaa(T59, X167, X168)) → split16_out_gaa(.(T57, .(T58, T59)), .(T57, X167), .(T58, X168))

The set Q consists of the following terms:

split16_in_gaa(x0)
U1_gaa(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.

(103) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.